The very bare minimum of latex to compile documents

Solution 1:

You could try one of the obsolete MSDOS distributions. Some of them are pretty small, like the base emtex (plus its fonts). From 1998.

Installation is documented in Norm Walsh's Making TeX Work, which is even more obsolete, from 1994, but it should be enough to tell you which bits of that you can throw away, conveniently summarised in the Making emTeX smaller section. He estimates that you can slim the distribution down to 5.5Mb.

But be warned: such a tiny distribution won't have staples such as pdftex, Metapost or TikZ, although there was already graphicx and Pstricks back in those days...

Cutting more means cutting fonts: maybe if you used Xetex's ability to support OS fonts, that would help? But I don't think emtex has Xetex...