Hydrogen donor without hydrogen acceptor?
As mentioned in the comments by Philipp, ammonium ions $\ce{R3NH+}$ are a good direct answer to what you are looking for.

The nitrogen in an amide $\ce{RCONHR'}$ is similar - the N lone pair is conjugated into the C=O bond and that makes it a very poor hydrogen bond acceptor, but the N-H lone pair can be a hydrogen bond donor. For more details, see this previous question with an excellent writeup by Jan. Now an amide can of course function as a hydrogen bond acceptor via the lone pairs on oxygen, so it's not strictly what you're looking for, but it was just something related. And it is of course directly relevant to the secondary structure of proteins.