Unique Passive – Insight: Gain ability power equal to 3% of your maximum mana. Unique Passive – Mana Charge: Grants +8 / 10 maximum mana (max +750 mana) for each spell cast and mana expenditure (occurs up to 2 times every 8 / 6 seconds).
By building a full set of items giving mana, (fully stacked Archangel's Staff, maxed Rod of Ages, Iceborn Gauntlet, Frozen Heart and Lich Bane) you will acquire total bonus 3150 mana, which will grant total 94.5 bonus ability power, not counting for champion's base mana rating and other stats given by those items indefinitely.
Archangel's Staff has great synergy with a Rabadon's Deathcap, as it increases ability power by 30%. This means that whenever your mana is increased, the Archangel's Staff will increase your ability power, and then the Rabadon's Deathcap will increase it again due to the 30% ability power passive.
To gain more AP from the Archangel's Staff than a single Rabadon's Deathcap (which gives 156 ability power if alone) a champion would need more than 3200 mana. However, a champion would only need 2000 mana for the Archangel's Staff to provide more ability power than a second Rabadon's Deathcap.
An early upgrade Archangel's Staff is comparable to buying a Needlessly Large Rod; the latter is somewhat more expensive (1980 as opposed to 1600) but you will usually gain slightly more ability power with Archangel's Staff, more mana regen and faster maximum mana building.
With the passive of this item, each 100 gold spent on mana at base efficiency will grant additional ability power worth 32.625 gold.
Note that most champions have around 800-1000 base mana at level 18.
Not counting manaless champions, Vayne and Ashe have 208 mana at level one, the lowest in the game, which combined with the staff's own 250 mana already equals 458. Taking into account the time (and therefore level) it takes to get the gold to buy it, it is practically impossible for this item to be gold inefficient.
Archangel's Staff grants 1 ability power per ~33 mana, giving a total of 67.5 ability power at zero stacks. It gives as much ability power per gold as a Needlessly Large Rod at 2500 mana.
The bonus mana does not stack with multiple Archangel's Staves, Tears of the Goddess or Manamunes. However, once an Archangel's Staff upgrades into Seraph's Embrace, the mana counter resets back to 0 and you can start charging another Tears of the Goddess or Archangel's Staff. The 1000 bonus mana from the upgrade will remain.
Assuming a charge is gained every 4 seconds, the 750 mana Unique Passive takes 6 minutes and 16 seconds to max. (The Tears of the Goddess takes 12 minutes and 32 seconds).
There is a 3 charge banking system to help burst casters.
Rammus'sPowerball adds one stack upon cast and another stack if ended early without a collision. Similarly, Janna'sHowling Gale adds one stack upon activation, and a second if activated early.
Singed'sPoison Trail adds one stack whenever the ability's toggle is active and the Tear/Archangel's Staff stack is not cooling down (i.e. continuously).