Tough call,
I have a bit of fond memories of my first computer I owned myself.. not shared with my siblings.. It was a Tandy Color Computer 2.
But I would have it was the first computer I built from parts was my AMD 386-40Mhz back in 1992, I spent an entire summer break from school doing odd jobs to pay for half of the parts and then Christmas money to pay for the rest of them.. Later I slowly added a sound card, one of those new fancy 2X CD-ROMs that cost $400.. the 4 Megabytes of ram that was about $500 as well. I mowed a lot of lawns and did a lot of painting to get that computer.
I'd say my runner up would be the computer I built after I got out of high school and when I landed a good Job, I built a Pentium II 400Mhz system when the processor just barely came out.. The processor alone cost me $1200. Lots of excellent games back then. Half-Life, Unreal Tournament, Warcraft III..Etc. That computer is still running good today for being 14 years old.