If you are looking to save money and go the easiest route, I would purchase a sleep mask.
Years ago I worked the night shift for a few years and couldn't find any curtain that would truly make it seem like "night time". This is the only thing that worked for me. It may take a few nights to get adjusted but there are so many out there, that I am sure you will find one that you like. Even today as I work a normal day shift, I still wear one at night because I am just to accustomed to it.
Build a frame from some lengths of wood, to fit in the window opening. Cover with a few sheets of paper, or some plywood. Maybe some rubber foam around the edges to close any remaining gaps. Add two handles to make it easy to move this frame.
Insert at night, remove in the morning. It's a bit more effort to build, but it's much faster to remove in the morning than anything taped to the window.
When I was a kid there was a bad heat wave and we used aluminum foil on the windows to keep the sun out. I don't know exactly what you mean by "do anything to my windows" but with a few cents worth of foil, you can make sure no light gets through.
There are large paper blackout shades (see example on Amazon) that block out all the light on the window's I've used them on.
They are a bit like giant sticky notes, with a sticky end that attached to the ceiling or window frame. During the day, when you want light, you use small plastic clips to gather up the extra paper material which lets light in. They cost about $30.
A couple of layers of newspaper attached to the window with masking tape.
Depending on how dark you want it, you can add fewer or more layers or leave smaller or bigger gaps.
Do not leave the tape on the window or window frame more than a few weeks, when the tape has been out for months it is very hard to remove.
This is an 'can use it already today' and 'more temporary' method than several of the others suggested here.
I also suggest a more long term solution, but it will only work in some cases.
If your window frame allows it you can also use an expandable shower curtain rod. For this you need a window that is set into the wall so you can use the telescopic and self-fixing nature of the rod.
If you can fit one of those you can hang two or three layers of shower curtain on it, maybe found in cheap or trift shops.
I had an eye surgery a while back and needed to black out a room under similar conditions. Though in my case it was the morning sun I was trying to black out.
I have tried 2 methods.
Do you care if its visible from the outside? I used tin foil taped directly on the window. Blacked out completely.
However, people will notice the crazy person with newspaper or tin foil on their windows.
Annoyingly, like Willeke says, the tape does leave behind residue that I had to scrap away with chemicals.
Also, on hotter days there was a fair amount of condensation between the foil and the window. Just a heads up nothing really to worry about there I believe.
I eventually made a shade the size of my whole window out of cardboard. The window was recessed so I would snug it against the top and bottom ledge, in front of the existing ineffective blinds. Blacked out completely.
The nice thing about this is that I could take it down and put it up easily. It left no trace of having been there. And I didnt look like a crazy person; at least from the outside. And it didnt cost me anything since I had cardboard lying around from various amazon packages. Maybe a few dollars in duct tape though.
If you want all the light-blocking of tin foil, and all of the subtlety of not having a shiny surface behind your window, consider "blackfoil" or "cinefoil." It's essentially aluminum foil, but with a matte black surface. It's generally used in theater lighting to block stray beams which might otherwise be escaping from the stage lights (or similar uses). On the order of $30 for a 50-foot roll.
If you don't need to be quite that thorough though (and you really don't), I had a similar problem sophomore year with construction lights, which I solved with a big roll of black bulletin-board paper and some tape. I picked mine up extra cheap at a going-out-of business sale, but even a full roll at full cost isn't that expensive (~$20). Maybe make friends with an art student?
If you haven't tried this before, do be a bit careful though. The combination of late nights and no circadian light source ended up being a bad combination for me. YMMV; but do monitor how it affects your wellbeing.
There's a product designed as a portable (suction-cup) blackout blind for travelling with babies. I've known it to be used by people living in furnished accomodation with rubbish curtains.
It's called the gro anywhere blind and is available from major internet reailers and shops specialising in prodcuts for babies. It's not the cheapest option but looks much smarter than tin foil or cardboard (from both sides).
Another option is a telescopic curtain pole which presses against the sides of the window recess. They're often sold for shower curtains but I've used them to avoid drilling into tiles in a bathroom. That link isn't the cheapest source (eBay/amazon etc. are worth a try) but should be quite stable as an example.
If it turns out that you are allowed to put up curtains, make sure you get blackout curtains. They really do what it says on the tin: we have white curtains on the balcony window in our bedroom, which faces south, and it's dark in the mornings thanks to the curtains.
I have used double layer of black garbage bags in the past. I've fixed the bags with black tape that doesn't stick much to avoid leaving marks. However, over time with many season change, this tape unglued by itself. I guess paint tape would be better for this, but I'm not sure it would not unglue over long period of time as well.
This is an alternative to the sleep mask because I have found that the sleep mask doesn't stay properly when I move and the elastic band are quite uncomfortable.
Note : When sleeping on your back, the nose act like a stopper to prevent the pillow from slipping down and obstruct the respiration path.
I assume that there are existing blinds/curtains? Hang a blackout sheets on them using clothes pegs.
One could also fit a baton using command-adhesive removable tabs above the head of the window so that it's not permanent, or wedge a baton inside the window reveal (cheeks of the window), just make it fractionally longer than the space between the reveals and spring it in place. card each end will protect the paint. Make sure you are friends with the maintenance guy for your block too!
Failing that, get a 750mw anti-light from eBay. Positioning needs to be exact, otherwise you will get standing bright spots. ;)