Just so you know, I take no credit for any of this, I mearly "improved" the previous design :P

 So we had a 1 bit adder. That overflowed pretty quickly. Then we had a 4 bit adder, which overflowed too. Then we had a 16 bit adder, but ultimately, it too overflowed. So, ladies and gentlemen, I took 4 16 bit adders and combined them to make a 64 bit full adder!

 Some might say the addition of a carry in and carry out input and output is entirely unnecessary, but trust me, but trust me, it's not.

 Now to make a 256 bit adder.

 p.s. This thing is so unnecessarily big, it actually overflows the integer limit of it's universe (i.e. my computer). It's literally breaking space-time!