It is possible to get a prepaid cellphone without a monthly fee. I would only recommend that to somebody who is going to use it very rarely, only for emergencies.
A typical deal might be that you pay $135, of which 35 is for the phone and 100 is money in advance for calls to be made over the next year.
If they year goes by and you don't make that many calls, under some plans you lose the left over money. Others let you keep it if you buy more calls for the next year.
These prepaid plans should have no monthly fee, which is what you apparently want.