It is theoretically possible to have a pi-bond to hydrogen, the necessary p-orbitals are there, but they are relatively high in energy. So, I would agree with jer83 that it is energetically unfavorable, although not impossible.
And there are examples of pi bonds without sigma bonds. I saw someone talk about a 4-membered ring system where there was evidence of an interaction of two p-orbitals on atoms at opposite corners of the ring. They referred to it as "banana bonding."