I can't think of any general-purpose architectures in common use which are really odd, but some DSPs have a 24-bit "byte". The microcontroller world is a bit more diverse too, especially with many of them being Harvard architectures (e.g. C compiles to 8051 with 24-bit big-endian pointers, and the various PICs have odd instruction-word sizes like 14 bits).
However, all of those still use 2's-complement integers. According to Wikipedia a few existing mainframes use 1's complement (CDC 3000/6000, UNIVAC 1100/2200).
I know the standard allows different bit representations, but practically speaking, today you're almost never going to find anything other than 2's complement integers and IEEE floating-point; and if you do happen to be working on one of those rare and unusual machines that don't, I think it'd be the least of your worries... even Linus says he doesn't care about Linux running on non-two's-complement or odd byte-size machines.
However, all of those still use 2's-complement integers. According to Wikipedia a few existing mainframes use 1's complement (CDC 3000/6000, UNIVAC 1100/2200).
I know the standard allows different bit representations, but practically speaking, today you're almost never going to find anything other than 2's complement integers and IEEE floating-point; and if you do happen to be working on one of those rare and unusual machines that don't, I think it'd be the least of your worries... even Linus says he doesn't care about Linux running on non-two's-complement or odd byte-size machines.