Claire Elisabeth- Beautiful! Can't go wrong with Claire Elisabeth.
Audrey Louisa- I don't like the "odd" sound in Audrey, but that's just me. Personally, I would prefer Aubrey Louisa.
Emma Catherine- Beautiful! I think Emily Catherine would also be gorgeous.
Louisa Madeline- Don't love the "wheeze" sound in Louisa, but once again, that's just me. Madeline Louisa would be my personal preference.
Hannah Eloise- Gorgeous! I feel the same way about Eloise as I do about Louisa. In this case, though, would Hannah Louise flow better?
Ainsley Grace- Adorable! Ainsley Grace is breathtakingly perfect. My vote goes to Ainsley Grace!
In the case of Hannah and Ainsley, I think they are such lovely names- no "Annie" needed. Good luck!
Hannah Eloise is by far my favorite! Personally I would keep it as Hannah rather than nn Annie, but that's me.
I like a lot of the others, but some don't sound as good with last name, i.e., Ainsley Egan/Audrey Egan (double ee/ee sound), and Claire Egan (just abrupt for my taste). None of these are terrible though. Audrey Louisa is really nice, and I also like Emma Catherine.