I think the on-chip bus interconnect and ALU are probably going to have some finite width, though. Above that number of bits in an operation, your instructions will get slower. Maybe that's what you have in mind, and you just want to abstract that away?
But I'm thinking more about having multiple logical CPUs, ALUs, or ports working on a single giant register. I think SIMD wastes a lot of hardware.
Example: radix sort needs to calculate a prefix sum. As the sum accumulates, it requires more bits, so the bit length should be adjustable. On x86, you can cast an array of bytes to a uint64_t and calculate eight prefix sums with a single multiplication. But eventually, you run into SWAR lane overflow problems. Ask an AI what this means and how the code works.
We need a CPU designed to handle lane overflow and dynamically adapt bit lengths, so at the start we do more parallel sums with less bits, and as the sum accumulate we increase the bit length gradually. This is just a simple use case. We could store more prefix sums as a periodic sum more deltas.
Besides a lower clock frequency, such a CPU would also have a power consumption, due to a greater total capacitance of the internal nodes.
The number of problems whose solution could be accelerated by such a structure is too low to make it worthwhile.
You'd essentially have a 1 bit processor with a cache that translated groups of bits to whatever memory access width was. CPUs already fill cache lines with transfers wider than registers, so the theory is there.
There's always a sweet spot for bang-for-buck on any tunable size like that. I doubt it is at 1-bit unless you can figure out how to do cache coherency with a few million cores.