My DS #2 is 16 months old and doesn't have a pillow in his cot.
But during day he lays his head on his Elmo pillow in lounge room on his fold out Thomas sofa
I don't remember what age I gave one to my DD, it wasn't quite that young but she was still in a cot, so could have been around 18 months? It is a normal adult one but reasonably thin/flat (which is what I prefer for myself anyway). She really doesn't care whether she has one or not though, and she often just moves down the bed so her head is nowhere near it anyway.
I'm one of those people that believe a lot of us adults only have neck/back issues due to using pillows as it puts us in an unatural position so to be honest I don't care whether she ever uses a big pillow!
DD2 is still in her cot and I haven't even thought about a pillow!!!
When my first son was born, the advice we were given is they only need a pillow when they are old enough to ask for it.
Like knockonwood said, it is better for their little bodies and spines to sleep flat.
Mr J (April 2005) Miss Z (Feb 2007) and Miss O (Oct 2010)