My parents bought my brother and I our first shared computer circa 1986. It was a great little machine called the Acorn Electron, with a mighty 32kb of RAM 
There's one for sale on e-Bay (current bid: £9.99) and I'm really tempted to bid for it!
Some of my favorite games included The Last Ninja (screen shot below), Starship Command and Barbarian. wikipedia has an extensive list of many of the games released for the Acorn Electron.
The Last Ninja contains a blend of exploration, puzzle solving and combat which is the sort of game I enjoy to this day! I really like Baldur's Gate, for instance...