The Z88 by Cambridge Research (the company Clive Sinclair set up after selling Sinclair to Amstrad) was a pretty impressive machine. It was actually the first properly affordable and truly portable computer; it is -according to legend- still very much in use today. The thing runs on 4 AA batteries for roughly 20 hours, sports a Z80A CPU and a multitasking OS, comes preloaded with quite a few programs, plays games and you can find out more about it via this article I wrote ages ago for Vintage Computing and Gaming.
Happily, grabbing one isn't too expensive either. Take this Z88 auction for example. It should end up costing less than the average contemporary game, and you'll be getting the fully working micro complete with its manual and carry-case. Seller ships worldwide.