Billy is a infant character who appears in the 1988 Disney/Pixar short, Tin Toy.
Even though he does not appear in Toy Story 3, a few of his toys do appear at Sunnyside Daycare. This may imply that he has become a lot older and outgrown his toys (including those not seen), and donated them to Sunnyside (we never even see what he looks like when he grows up).
In the credits of Tin Toy, Billy only plays himself.
In Toy Story 2, when Hamm is rapidly turning the channel, Billy is briefly shown between channels.