The first, really small working computer (several years, note, ahead of the MacMini) was probably the Espresso, made by CappuccinoPC. Now it has rivals. Most of these are desktop versions of Netbooks such as the Asus Eee Box.
But there is also the VIA Artigo Pico-ITX which can fit in the spare drive bay of a standard beige box. Should you, er, feel the need.
Whatever. The Artigo Pico-ITX needs quite a few things to get it working such as hard drives, memory and an operating system. Plus the stuff you never see in any tiny-computer advertisment, a power supply and connecting cables. Pish, you say. But count the cables for boring old things like keyboard, mouse, power, screen, webcam, speakers, ethernet and the odd USB memory stick. The box is almost lost in this ratsnest of disparate cabling.
So back to the Eee Box. At least you can bolt it on the back of your old flatscreen monitor. Doesn't stop the ratsnest but at least you can stuff it all down behind the desk.