What is another word for big fan?

It depends. If you were saying, I am a big fan of Daniel Craig, you could say, I idolise, Daniel Craig.

Followers and Admirers:

Stalker, addict, admirer, aficionado, devotee, follower, freak, groupie, lover, supporter, zealot

Air and Cooling Devices

If you mean a large air circulation device, you could mean: blower, chiller, HVAC System, radiator fan, etc.