No damper at all, ouch!
There are inflatable balloon fireplace dampers that are made to fit your fireplace exactly for around $50. They plug the chimney flue like a metal damper would and you can reuse it if you want to have a fire in the fireplace.
When I do home energy audits and the chimney is found to be a major air infiltration spot, I generally recommend for them to use a chimney damper balloon to stop the heat loss.
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Just buy a a folding glass fireplace door, it will stop or slow heat from escaping.
after you finish burning your wood, but a piece of plywood inside the cimmeny
i am not a fireplace expert, give me more details on what a fireplace damper does!