I wrote a paper on three-valued logic for a math course - does that count? It was only about algorithms, however, not hardware.
Historically, there
have been base-10 computers, particularly mechanical ones, and computers with word sizes that were not multiples of 2, although you don't see them around much anymore. Survival of the fittest.
For his landmark book series
The Art of Computer Programming, Donald Knuth invented a mythical computer named MIX (model number 1009 based on the interpretation of M I X as roman numerals). The MIX machine is programmed in assembly language and emulators have been written for all types of real computers. The remarkable thing about MIX is that the machine can be either binary or decimal, and the programmer doesn't know or care! In other words, an assembly language program is written that will run properly whether the program is run on a base 2 implementation or on a base 10 implementation. Ternary (base 3) implementations are even possible!
John Atanasoff gets some of the credit for realizing that computers would best operate in base 2. Remembering the 1937 project where he designed one of the first electronic computers on a cocktail napkin, he said "It was at an evening of bourbon and 100 mph car rides when the concept came, for an electronically operated machine, that would use base-two (binary) numbers instead of the traditional base-10 numbers, condensers for memory, and a regenerative process to preclude loss of memory from electrical failure."
So perhaps we're all using binary computers only because of a drunken speeder's fantasties!