I just bought a home recently and there is no damper to close off the
chimney. There is a (bird) cap on top, but I want to close off the
whole thing because I don't want my cats climbing up in it.

I need a very cheap and very easy suggestion on how to fix this. I
have considered fireplace doors, but can't afford them right now. I
have no money and no expertise in home improvement.

I just stuffed a folded fibreglas blanket up the flue, so that it fit
tightly. Big box stores sell 4" and 6" batts or rolls of this stuff for
wall insulation. Get the smallest amount you can buy, and preferably
unlined, with no paper backing. You can trim to size with a large pair of
scissors. This will stop all drafts, and may even discourage your cats. Here
is a picture of the stuff:
http://www.littlebrothers.org/hought...late/batts.jpg