I've been wondering this myself for ages too
I guess the basic ones just have a simple ASIC with a 4 bit core that's highly optimized for calculator operations. Certainly nothing general like a PIC or AVR, maybe something highly obscure like from NEC.
For fun, a while ago I designed my own four bit core to make a simple four function calculator and wrote the code for it. An interesting trade off is whether to include a special instruction to get something done or use multiple, more simple instructions but use more ROM. I don't know what's cheaper in silicon, logic or ROM.
Currently it's only running on a simulator on a PC, but I still plan to create an actual hardware version, with an AVR emulating this P-Code (actually, the first calculator using the 4004 used P-Code
), the challenge to keep everything within 2K of Flash.
edit:
One even more totally outrageous thing I did a couple of years ago, was reimplementing the original HP-35 with a FPGA, from the descriptions and code from the original patents which are very detailed. Unfortunately, I never got round to actually finishing it