This has been inspired by this thread:
http://forum.square-enix.com/ffxiv/threads/9521-Count-me-amoung-the-skeptics-...
but it addresses a number of topics so I thought I would post it separately in here to discuss it more specifically.

What would you think of quests that would allow to replenish animas?

If it's well designed and balanced, it could be a rallying type of content that would make you accomplish something instead of waiting for them to add up with time and running around for long periods between camps and cities.

The thread I linked also proposed quests to raise anima cap, which I think would be a great idea, as long as you need to accomplish quite a few deeds in Eorzea before having access to that content.