As the title suggests, can a replicator replicate a (presumably slightly smaller)replicator?
If it could, could it replicate a better replicator? One that works at the quantum level, like the transporters?
I remember reading in one of the books related to the earlier motion pictures that it was (of course) Mr Spock who first came up with the concept that eventually became the replicator. He figured out how to use the Enterprise's transporters to beam up many more supplies than the ship could actually fit in it's holds, but each transport was aborted in such a way that it could be resumed later. This was the precursor to what eventually became the replicator.1 Supposedly this really taxed the Enterprise's engines, to produce the energy needed to sustain all those transports. Using this process, a replicator could only replicate itself if another replicator were waiting.
From here, we can take a mental leap to infer that later replicators evolved a bit. Presumably, someone figured out that they could extract and digitally store just the pattern data from a transportation, without needing to sustain the actual transport process, and also devised a way to use the stored patterns to "finish" a transport process from stored bulk matter on demand, such that anything that had been "transported" once could be reproduced. So a replicator is just a follow-on to transporter technology.
If this is an accurate description of Star Trek replicators, then a replicator can replicate itself as long as it has a digital pattern and a source of the right bulk material to work from.
1Unfortunately, I have no chance of finding a reference for that now.