Small apartment with 8 single-pane windows with wooden frames that are not winter-friendly. The place is rented so can't replace or do anything major. Already filled in gaps in frames to stop drafts. Is there anything at all that can be done to the windows themselves to help improve the situation?
Simple solution would be curtains, but they block the light during the day.
More complex solution would be acrylic glass.
Since you can not do modifications to the windows get some acrylic glass (plastic) cut to the dimensions of the current glass. Then just mount it on top of the glass. It comes in 4x8 foot panels, so lay it out to get maximum cuts, since it is not cheep. It is flexible light weight and easy to cut.
You can find it in Home Depo and others.
Here is an example of my patio window with acrylic glass (4X8) that I installed 5 Yr ago, and still looks great.
Curtains are your friend, the heavier the better. Although be weary of blackout curtains. Make sure you check the packaging for any kind of "reflective cooling technology" or similar wording.
Buy some high quality reflective film, make sure to check the packaging to ensure it's multi-seasonal. The current film on my windwows drops the temperature of the glass by 6 degrees in the summer and raises it 3 degrees in the winter.
As Ruskes said in his answer, acrylic glass can help out a lot, it's terribly expensive nowadays, so if you've got a second-hand/discount hardware or construction store, keep an eye out for undamaged sheets. And be very careful while installing... If your cut is too large, don't try to force it. Chips on your edges will grow.
It's also worth mentioning working shutters. Every layer of insulation you can put between you and the outside helps. Not sure if that's too much for a rental though.