WordPress is a constantly evolving well-oiled machine. If you stroll through the Codex you will see the best documentation ever of the evolution of a piece of software. Lots of functions and template tags have been deprecated over time. Do they break immediately? No – but eventually it’s bound to happen. Plus, if you are using plugins anywhere, you’ll have to upgrade those – and if you’re not upgrading your WP then that will cause a problem. If you don’t upgrade the plugins, that will cause a problem.
Furthermore, look through the history of WP upgrades and their reasons. A number have been for security holes. The farther out of date, the bigger risk you have of someone sabotaging your site.
There are 1,001 reasons you need to keep your WP installation up to date… I can go on if you like… ??
Furthermore, with one-click upgrading, there’s just no reason not to (but take it from someone who’s learned… you have to backup your database, and preferably your site files too, before doing an upgrade! Every time!)
Just trying to pass on some wisdom I’ve learned over time…