Archangel's Staff has great synergy with a Rabadon's Deathcap, as it increases ability power by 35%. 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 35% increased ability power passive.
This is equal to 4.05% of the holder maximum mana.
Note that most champions have around 800-1100 base mana at level 18.
Not counting manaless champions, Kalista and Vayne have 232 mana at level one, the lowest in the game, which combined with the staff's own 250 mana already equals 482. Taking into account the time (and therefore level) it takes to get the gold to buy it, it is almost impossible for this item to be gold inefficient.
Archangel's Staff grants 1 ability power per ~33 mana, giving a total of 87.5 ability power by itself at zero stacks.
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 2 charge banking system to help burst casters.
Rammus'Powerball 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).