Kings Landing

Because Chris asked and I realized I had actually never  thought about the dragons... 

The Dragonpit is actually a huge, cavernous building on Rhaenys' Hill. The doors are solid bronze and the roof is domed. Within are the three current Targaryen dragons: one green, one white, one black. The dragons themselves are about a year and some change old, so they're about the size of a St Bernard. Aenyris and Jaehaerys spend off-camera time with them. The doors to the dragonpit are opened during the day so the dragons can fly around a bit- they are a common enough sight in the sky over the harbor, and there's whispers that the green or the black one may have eaten an unwary sailor or two but no one seems to have known the sailors personally. 

Aenyris is mostly found by the dragonpit because she doesn't have to do monarch-y things so she's got more time to work with them. There are goldcloaks posted at all times. If Aenyris is there and she approves of you, she'll call out the dragons so that you can see them. Usually the white or the black since the green one is actually the more temperamental. 

And since Aenyris isn't played, just go ahead and assume if you visit the Dragonpit during the day you'll find her.