There are different free alternative service available.
Windows Remote Desktop
Available for Windows and Mac.
Software from Cisco.
It is free for the client in the computer remotely access, but you need a license on the computer performing the connection.
Theorically it is possible to control remote computer using WebRTC on supported browsers (Firefox, Chrome, Opera), though I have not experimented myself.
- “5 alternatives to LogMeIn Free for remote PC access” by PCWorld.com