Disclaimer[]
This is incredibly outdated. I think the last time I updated this was S1. Don't refer to this.
Intro[]
So I've been meaning to do a couple comprehensive guides for my own sake as well as the community's, and I thought I'd start with this one, as it should be relatively simple and objective and it particularly interests me, so here we go.
What is cost effectiveness?
Cost effectiveness is the measure of how much you're getting for your gold when you buy an item. It allows you to know how "good of a deal", so to speak, the item is, and should be one of the things taken into account when choosing an item to build. It takes precedence over early game as more cost effective items will allow you to get an early advantage, but is of a bit less importance once your slots start filling with items and slot effectiveness becomes important. It's something that's rarely adressed but is of high importance, so I thought I'd try to make a comprehensive guide.
Cost Effectiveness by Statistic[]
This section is less useful than my planned cost effectiveness index section I plan to make. I may delete it later but for now it's useful for keeping track of initial stats and if someone happened to be trying to increase a single one I suppose it could be useful.
Attack Damage: Compare to 
B. F. Sword(36.7 gold per AD)[]
Aegis of the Legion 8 AD(from the aura)/1925 gold: 240 gold per AD
Analysis: I wouldn't buy this based on the AD. It's more cost effective if your teammates use the aura (you can effectively get it down to 48 gold per AD if your whole team uses the AD from the aura), but you should buy it more for the defensive stats it provides. If you want an offensive aura to synergize with AD, get 
Stark's Fervor.
Atma's Impaler 2% of max health as AD/2405 gold: 2405 gold (divided by every 50 hp you have) per AD
Analysis: Assuming 2000 health, Atma's has a cost effectiveness of 60.1 gold per AD. However, past 3250 health, Atma's becomes more cost effective than a B. F. Sword, and has some other great stats to boot. Great if you can build around it with items like 
Frozen Mallet, 
Banshee's Veil, and 
Sunfire Cape.
B. F. Sword 45 AD/1650 gold: 36.7 gold per AD
Analysis: It's exactly as cost effective as a B. F. Sword. Big surprise there.
Bilgewater Cutlass 35 AD/1825 gold: 52.1 gold per AD
Analysis: Decent cost effectiveness for AD. Better when you consider the lifesteal and the active.
Doran's Blade 10 AD/475 gold: 47.5 gold per AD
Analysis: Very cost effective considering its other stats, but terribly slot ineffective. Stacking Doran's items is a good early-game tactic, but your inventory fills up fast and you'll quickly have to sell them as they provide small stat increases that aren't useful later game.
Frozen Mallet 20 AD/3250 gold: 162.5 gold per AD
Analysis: Get it for the slow and to synergize with Atma's Impaler (it then adds 14 more AD for a cost effectiveness of 95.6 gold per AD, which is still pretty bad, but the slow is awesome). Don't expect it to cover your AD needs.
Guinsoo's Rageblade 35 AD/2235 gold: 63.8 gold per AD
Analysis: Rageblade is incredibly cost effective. It gives a lot of great stats for very very cheap if you can keep up your stacks by autoattacking or spellcasting. The only issue is that not everyone can make use of those stats, as many autoattackers have very little use for AP and few AP-users autoattack regularly. If you use both well, however, this item is great. If you're playing a champ that benefits from AP and autoattacks often (
Twisted Fate, 
Rumble, and 
Kayle are a few good examples), try it out.
Hexdrinker 35 AD/1800 gold: 51.4 gold per AD
Analysis: Surprisingly cost effective for AD alone for an item that gives 35 MR and a magic-damage shield. Great if you're having trouble with magic damage.
Hextech Gunblade 60 AD/3625 gold: 60.4 gold per AD
Analysis: Again, surprisingly good considering its other stats, but I wouldn't get Gunblade unless the champ I was playing could benefit at least from the spell vamp it provides. If you can benefit from the AP and the spellvamp as well as the AD and lifesteal, again, this is an awesome item.
Infinity Edge 80 AD/3830 gold: 47.9 gold per AD
Analysis: I'm a bit surprised at this one... it's about as cost effective for AD as a Doran's Blade and provides good crit chance and a good passive. I still don't know about stacking them, but this surprised me.
Last Whisper 40 AD/2290 gold: 57.25 gold per AD
Analysis: not the greatest cost effectiveness for AD alone, but if your opponents are stacking armor this item will provide more dps than any other for physical damage champs due to its passive, which essentially makes all your AD more effective. Great item, but situational.
Long Sword 10 AD/415 gold: 41.5 gold per AD
Analysis: Better for AD alone than a Doran's Blade. Good for early game but less cost effective than a BF sword. I would pass it up for a doran's blade if you for some reason aren't planning to build it into anything, though.
Madred's Bloodrazor 30 AD/3800 gold: 126.7 gold per AD
Analysis: pretty terrible when you look at AD alone. It provides a bit of armor too, but atma's is cheaper and provides more armor and usually more AD. Madred's Bloodrazor has decent attack speed and a great passive going for it too, but I can't begin to understand why so many people rush this item or use it as a core item. If they're jungling I can sort of understand it but otherwise it's pretty expensive and pretty bad early game unless you're against someone who rushes 
Warmog's Armor for some reason. That being said, it allows autoattackers to deal a lot of magic damage late game, so it can be good if your opponents are stacking up on armor and health and ignoring magic resist, but that shouldn't really be happening.
Madred's Razors 15 AD/1000 gold: 66.7 gold per AD
Analysis: Twice as cost effective as Madred's Bloodrazor for AD, just to point that out. Only get it to speed up jungling, though, as it's pretty bad when you don't use the passive frequently.
Manamune (20 AD + 1 per 50 mana)/2110 gold: 105.5 gold per AD on manaless, 35.2 gold per AD with 2000 mana
Analysis: Yep, that's the reason not to get this on manaless champs or champs that can't come close to maxing it. Maxed, the mana from this item alone gives 27 more AD for a cost effectiveness of 44.9 gold per AD, which is great, and that's not even including base mana or any other bonuses you may have (at 1900 mana, which is easily attained with Manamune if you can max it, it becomes more cost effective than a B. F. Sword). Great raw damage item if you can max it and it tends to solve any mana problems you might have. Note however that both passives are unique (unlike 
Archangel's Staff and stacking them is a pretty bad idea.
Pickaxe 25 AD/975 gold: 39 gold per AD
Analysis: Obviously pretty cost effective for AD since that's all it provides. It builds into some good items and can give a better boost early game than a Long Sword.
Phage 18 AD/1315 gold: 73.1 gold per AD
Analysis: Just like frozen mallet, you pay a lot for the slow and the health. Mostly the slow; so don't get it if you don't need it. Phage isn't great unless you intend to build it into something else, either, so I wouldn't get it without the intention to build it into trinity force or frozen mallet.
Sword of the Occult 10 AD + 5 damage per stack/1369 gold: 136.9 gold per AD
Analysis: It goes without saying that you need stacks for this to be anywhere near cost efficient. At 6 stacks it exceeds the efficiency of a B.F. sword at 34.2 gold per AD. This is essentially the breaking point. If you aren't doing well enough to maintain 6 stacks (which you can maintain if you get 1 kill or 2 assists for each death) then you should get a B. F. Sword instead.
The Black Cleaver 55 AD/3065 gold: 55.7 gold per AD
Analysis: Surprisingly good for AD alone for an item that gives attack speed and up to 45 armor reduction. Between those 3 stats BC should turn out to be incredibly cost effective for increasing dps on an autoattacker.
The Bloodthirster(stackless) 60 AD/3200 gold: 53.3 gold per AD
Analysis: Decent for increasing AD without stacks, but considering all it gives other than AD is lifesteal, it needs stacks to really be worth it.
The Bloodthirster(fully stacked) 100 AD/3200 gold: 32 gold per AD
Analysis: Very cost effective for AD alone, and it gives great lifesteal as well. It's a good candidate for upgrading that B. F. Sword into, but I guess so far this has shown that the other two are as well. If you can farm 40 minion kills quickly and can stay alive well enough to keep the stacks reliably, The Bloodthirster is arguably the best item for increasing AD (although The Black Cleaver is still probably better for increasing dps).
The Brutalizer 25 AD/1337 gold: 53.5 gold per AD
Analysis: Decent cost effectiveness for AD alone. Good early armor penetration. Good early cdr. Builds into 
Youmuu's Ghostblade. What's not to love?
Tiamat 50 AD/2070 gold: 41.4 gold per AD
Analysis: more cost effective than a Doran's Blade for AD. That's a bit surprising. It also provides some mana and health regen and can help with farming if that's an issue. Rarely should you get this, but interesting nonetheless.
Trinity Force 30 AD/4070 gold: 135.7 gold per AD
Analysis: It really shouldn't be surprising considering how many stats and effects Trinity Force provides. It's obviously not going to be cost effective for AD alone, but it's not like this item isn't valuable.
Wriggle's Lantern 23 AD/1600 gold: 69.6 gold per AD
Analysis: Wriggle's is a package deal. It provides armor, lifesteal, and a passive that speeds up jungling in addition to its AD. Not all that attractive when you're not on a jungler, but the armor and lifesteal is enough to make this a great item.
Youmuu's Ghostblade 30 AD/2687 gold: 89.6 gold per AD
Analysis: Bad for AD alone. Great nonetheless for any melee champ provided you use the active at the right time, however.
Elixir of Fortitude 10 AD/250 gold: 25 gold per AD
Analysis: Most cost-effective AD boost in the game. The only issue is that it's temporary and it doesn't stack.
Ability Power: Compare to 
Needlessly Large Rod (20 gold per AP)[]
Abyssal Scepter 70 AP/2650 gold: 37.9 gold per AP
Analysis: This is one of my favorite items. MR reduction, AP, and MR. It greatly increases your dps while making you tankier. It's not that great for AP alone, but like black cleaver it increases dps in more ways than one.
Amplifying Tome 20 AP/435 gold: 21.8 gold per AP
Analysis: Really shouldn't need one.
Archangel's Staff 45 + 3%/max mana per AP/2855 gold: 63.44 gold per AP on manaless, 27.19 gold per AP with 2000 mana
Analysis: Again, this is why you don't get it on manaless champs. This requires more mana than Manamune to be cost effective, but that's fine, since anyone that benefits from it will be both more likely to be able to fill it, more likely to already have a higher base mana pool, and more likely to buy other mana items such as 
Rod of Ages.
Blasting Wand 40 AP/860 gold: 21.5 gold per AP
Analysis: The cost effectiveness difference between Amplifying Tome and Blasting Wand is almost negligible; if building something like 
Rylai's Crystal Scepter I don't think it'd be a terrible idea to start with the Amplifying Tome if you didn't have enough for the Blasting Wand yet.
Deathfire Grasp 60 AP/2610 gold: 43.5 gold per AP
Analysis: Over half the gold spent on this item is spent on other things (including the active; don't forget to use it). It's a solid item and great against health-stacking champs, but it's one you should be getting less for the AP alone and more for its entire kit.
Doran's Ring 15 AP/475 gold: 31.6667 gold per AP
Analysis: An early-game item. If you just want AP, you're better off starting with an Amplifying Tome and building it into something else, but then you're missing out on the mana regen and health provided by this item. Get it if you could use those stats (and there are few mana casters who couldn't).
Fiendish Codex 30 AP/1245 gold: 41.5 gold per AP
Analysis: 2 of these will provide 4 more MP5 and the same amount of AP as a Deathfire Grasp for 120 less gold, but you'll miss out on 5% more CDR and the active. It might be worth stacking them if you planned to build 2 of the items they build into and needed the extra mana regen fast, but the wasted CDR is kind of problematic.
Guinsoo's Rageblade 45 + 6 per stack AP/2235 gold: 49.6667 gold per AP at 0 stacks, 24.03 gold per AP at 8 stacks.
Analysis: close to cost effective for AP alone, but don't get this if you don't autoattack. Inversely, you shouldn't get it if you can't use the AP either. It's amazing for anyone that can keep up the stacks and benefits from both.
Analysis:
I'll continue this later. It's taking a bit longer than I expected :P.