I think it would be more tricky to find out what bands have had the least member changes.
ZZ Top
Rush. One member change. Can't come up with any with zero.
Obviously, any RECENT band would be more likely. If they've only had one album, have only been around for less than, say, 3 to 5 years, chances are, they haven't had any line-up changes since their inception.
Also, band-name changes could change. I mean, Dream Theater was once Majesty, but that changed when Charlie Dominici joined the band and replaced Chris Collins, so in reality, Majesty really only had one line-up...kind of.
Anyway, I think if you're going to look for a band with the fewest line-up changes, I'd say they'd have to be around for at LEAST 5 years (10 or more would be impressive), and have at least a few albums under their belt.
-Marc.