My first was an 8080 machine I wire-wrapped, loosely based on a series of Popular Electronics articles. It had 1K SRAM, a hex keyboard salvaged from calculator parts and a 4-digit, 7-segment display. I've since re-discovered the article series and have some of my original design notes. I may try to recreate it.
I still have 7 of the RAM chips (2114's) and only recently lost the EPROM -- I think it fell into the trash.
Then my dad got a ZX-81 and shortly thereafter, I got my TI-99/4A. Man, I loved that TI! Had the expansion chassis, RAM card, floppy drive, cassette drive, speech synthesizer, color monitor, etc. I eventually traded it for a 1975 Chevy Impala. I shouldn't have done that.
But I still have my dad's ZX-81.