How to run a program with administrator privileges on user login under Windows?

The best way (and the only simple way if you not only want this to run as admin but also without an UAC prompt) to do this is by creating a scheduled task. There you can specify which user you want to run it under but also that it should "run with highest privileges". This is probably what you really want because the checkbox button "Run as administrator" does trigger a UAC prompt rather than really using an administrative user. If you also specify a trigger as "on user logon" it should have the same effect as using the startup or run key.

Alternatively you can use runas with the /savecred option to store the password of another user once and have it cached forever but it does not work around UAC afaik.