Well, first I bought an abacus for 3 denarii. It was a bit slow so I bought some grease and oiled its runners - my first overclock I suppose. Its reliance on base 10 arithmetic was problematic though for working in Roman numerals in the market, so I was forced to upgrade. A friendly salesman from Rhodes sold me a wax-tablet and an Epson stylus with which to write: this was a massive improvement as floating-points were more easily represented than with the abacus. Its advanced Non-Volatile technology meant that its memory's contents were not wiped each time I left it off overnight. It was also pretty crash-resilient, although high temperatures caused data corruption (i.e. wax melting) just as they do today.
No, really, my first proper computer was Babbage's Arithmetical Engine. Wasn't exactly portable, but it did its job magnificently. It was only replaced when Commodore released the Amiga 500+, the biggest, badest-ass mofo going at the time. Motorola 68000, multi-way custom chipset (ECS), 1 Mb chip RAM (24 bit address space though), KickStart 2.04, 880K 3.5" floppy. Kicked the PC-of-the-time's butt with its 4096 simultaneous colour-palette, 1280x512 (+ a bit of overscan) screen resolution, AutoConfig device controller (bettered only by very recent BIOSes), pre-emptive multitasking scheduler, 22 kHz 8-bit stereo sound, Unix-oid operating system. Only cost £399, a tiny fraction of the price of substantially worse PCs. I kept my Amiga (and its bigger brother, an 68040 A4000 that triple-booted WB 3.0, MacOS 8.1, and NetBSD 1.3) until 1998. It took until Windows 2000 until I actually preferred my PC.