G = Gp{ x,y,z | [x,[x,y]]=z, [y,[y,z]]=x, [z,[z,x]]=y }
and it was proposed some time ago as a possible example of a finite group with balanced presentation. It was known to have a finite quotient of order 2^24*60, but nothing else was known about it.
KBMAG proves it automatic in about 20 minutes, and from the output automata it is apparent that one of the generators has infinite order. Thus the group is infinite.