OS X has VNC built into it. Enable screen sharing (Sys Prefs > Sharing > Screen Sharing) on the machine you wish to control.
From the other machine, just type vnc://<ip_address> into Safari and it'll load over the internet (or local network if you have more than one machine on the same LAN).
You may need some port forwarding trickery on the router, mind, so you could consider this solution a little more advanced than the simplicity of LogMeIn.
I use LMI for 5 or 6 machines, works flawlessly, even have the universal iPhone/iPad app. May seem pricey, but it's just one of those apps that are so well developed they're worth their weight in gold and you'd wonder how you ever did without.
Ironically, LME is on sale in the app store until January 4th. It's currently £12 and opposed to £18.