I was writing up an algorithm that involved knowing the size of integers my hardware can manage without having to resort to software implementations of math operations and the additional computational load that comes with them, and I realized that I've never given this much thought.
In theory, a 64-bit computer has 64 lines on the address/data bus, and thus can't send an integer to the CPU's ALU any higher than 2^64-1, making that the starting point. But I also know that some modern hardware has support for SIMD operations with 256-bit data, or even higher, and how that's possible I don't know (multiple operations to load in the data, 64 bits at a time, I assume?).
Without limiting this question to specific hardware, which would greatly diminish its usefulness, is there a reasonable rule of thumb for how large of an integer a typical modern computer (2015-2020 era) can handle in a single go?