Why does my aiptasia keep coming back?
Usually because a piece got left behind. Aiptasia can regrow from a tiny bit of foot left on the rock, and there are almost always more hiding in the rockwork than you can see. New ones can also ride in on new rock or coral.
We watch aiptasia come back every day (on purpose)
We grow aiptasia to feed our berghia, and the way we multiply it is by clipping the heads off. The stalk grows a new head and the clipped head reattaches somewhere new. If you've been cutting, scraping or picking at yours, you may have been running a tiny aiptasia farm without meaning to.
Clipping the head off makes two (starts at 2:23)
Where the new ones come from
The biggest one is pieces left behind. Any little bit of foot left on the rock can grow into a whole new aiptasia. The next is the ones you can't see. Aiptasia love the cracks and back sides of rockwork, so the ones you spot are rarely all of them. And new rock and coral frags can bring hitchhikers with them, which is how a lot of people end up with aiptasia in the first place.
A lot of reefers also say a stressed or dying aiptasia can release larvae. We haven't seen that ourselves, but it's one more reason picking at them doesn't seem to help.
What finally stopped it for us
In our 16 gallon, we added four berghia to about six aiptasia. They were gone in about a week, and we haven't seen one in that tank since. Berghia hunt down the hidden ones you can't reach, which is the part scraping and cutting can't do. Here's how to get rid of aiptasia (what we tried, what we skipped, and what worked).