The Usborne Touch and Feel books were also very popular between 1 and 3. Lily's favorites were That's Not My Puppy, That's Not My Train, and That's Not My Snowman.
The Hinkler Books' Lift-the-Flap books also lasted us between 1 and 3. Lily loved Words and Animals, but Farm was her favorite.
Lily received David McKee's Elmer as a baby shower present, and we continue to love this book and the rest of the series. Elmer is a patchwork elephant who doesn't like being different from the other elephants but learns to love who he is. Great for any age!
Other books we read for the first time between 1 and 2 and still read today at almost 4 are Brown Bear, Brown Bear What Do You See by Bill Martin and Eric Carle and Eileen Christelow's Five Little Monkey's Jumping on the Bed (and the rest of this series).