I know this question is fairy broad and subjective, but I'm wondering if there's been any published research into what an optimal number of achievements is and what kind of challenge they should present. The game this question directly relates to is a shoot-em-up, but an ideal answer is fairly theoretical.

If there are there are too few achievements, or they are not challenging, I would expect they would fail in their goal to keep people playing. If there are too many, or they are unreasonably difficult, I would expect people to quickly give up. I personally witnessed the latter happening in Starcraft 2; a section of the achievements would have you win hundreds of games against their AI opponents (boring!)