Seeing as using technology isn't a stereotypically babyish thing to do, I only recently looked into what apps there might be aimed at babies/toddlers, and there's actually plenty of really good ones (as well as some very dull ones!).
Also, I like the fact that you can plug your headphones in, and be on your iPad/iPhone/etc. doing Little things, and nobody around you will necessarily be aware of it. So, who else likes babyish apps? Here are a few I find particularly fun:
In the Night Garden - Really fun, though I'm biased because it's my #1 favourite little show.
Peekaboo Barn - Lots of fun naming animals and counting. Plus there's a free version of it.
Wheels on the Bus - fun to sing along to, and you can record your own voice on it. I like doing this with Mommy so she can help me 'learn' the words and sing it with her.