Heck, even outside the realm of embedded applications, the Atari 2600 only had 128 bytes of RAM, and look what clever programmers were able to accomplish with that.

I believe some PICs and AVRs have as little as 32 bytes of "RAM" (actually no RAM at all, its all registers)

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Want to design a PCB for your project? I strongly recommend KiCad. Its free, its multiplatform, and its easy to learn!
Also, I maintain KiCad libraries of Retro Computing and Arduino components you might find useful.

I'm not familiar with the AVRs; but the smallest PIC I've used, the 8-pin PIC12C673, had 128 bytes, mapped the same way as the special-functions registers (FSRs), 96 of those RAM bytes being in bank 0 and the rest in bank 1. You could have an instruction MOVF Foobar, W for example, to copy variable Foobar in RAM to the W (ie, working) register of the processor. W is like the 6502's accumulator, but these smaller PICs don't have index registers or any other processor registers that are directly workable by the programmer.

I have not looked at the 6-pin PIC10's but the smallest amount of RAM I'm aware of having been offered in a PIC is the 25 bytes of the PIC12C508 which works basically like the above. The '509 had 41 bytes. The 508's RAM was at addresses 07 to 1Fh in bank 0 (while addresses 00-06 were SFRs), and the 16 extra bytes on the '509 were at addresses 30h-3Fh in bank 1. Managing the RAM banking (and paging in program memory if you have to do that too) dramatically reduces efficiency; so you have to plan your usage judiciously.

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http://WilsonMinesCo.com/ lots of 6502 resources
The "second front page" is http://wilsonminesco.com/links.html .
What's an additional VIA among friends, anyhow?

Atmel's first the AT90S1200 was a RAM-less device. Having 32 8bit registers, 1K ROM, and 32 (or 64?) bytes of EEPROM it was a neat solution (20 pin only) for some applications especially where saving a setup (=> EEPROM no batteries) was necessary or sophisticated. And it was in system programmable - far superior than its competitors.

RCA's early CMOS CPU, the COSMAC aka 1802, was similarly well-endowed with registers - it has 16 two-byte registers. Might be enough for point-of-sale or calculator application. (Conversely, as it has built-in single step/DMA mode, it can also be run without ROM, because it's unusually easy to setup a front panel to toggle in a bootstrap.)

The biggest 8-bit PIC has 3.5K of RAM (for USB buffering). Most of them have a lot less. If you used a 65c134 or 65c265 for a simple control application then you could probably get away with no additional SRAM just a ROM for your code.

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Andrew Jacobs
6502 & PIC Stuff - http://www.obelisk.me.uk/
Cross-Platform 6502/65C02/65816 Macro Assembler - http://www.obelisk.me.uk/dev65/
Open Source Projects - https://github.com/andrew-jacobs