The size of a register is not the largest value you can conveniently count on a computer. You can use multiple registers.

Old computers often had a "carry flag" specifically to make this easier e.g.on Arm:

    add r0,r0,#1
    adc r1,r1,#0

But even on RISC-V, often criticised for not having a carry flag, it's not hard:

    addi  a0,a0,1
    sltiu t1,a0,1 # set to 1 if a0 wrapped back to 0
    add   a1,a1,t1