Esoteric processor
Esoteric processors may exist only as thought experiments, or perhaps experimental processors could be counted as esoteric. If it is useful and in some sense or in some kind of computation better than previous processors, it probably can't be classified as esoteric. It could mean 2 things: It is (1) optimized to run some esoteric programming language or (2) implemented in some strange way. For every esolang, there is theoretically an esoteric processor.
In case 2, what features could an esoteric processor have? Trinary numbers, analog computing instructions, physical random number generator, instruction for division by some arbitrary number(7,13,100,12345), asynchronous operation, 7-bit bytes, 25-bit numbers, adjustable length numbers, etc.
A third definition could be that it is designed to run an esoteric operating system, whatever that is. Then there would be an esoteric computer.
Possible esoteric processor: Processor/1 ??