Connecting to Cyberspace From a Pentium MMX (Part 1)

So there is this Guild Competition in the Technica Guild, of accessing Cyberspace on the Crappiest hardware. The current record holder, at the time of writing, is a Raspberry Pi 2B. When I saw that I got wondering instantly, could I do better?

I have some Allwinner F1C100s and F1C200s boards around with 32 and 64 MB RAM, and a single ARMv5 core at 533MHz,... but that's just not...