The version of Rogue from the package "bsdgames-nonfree" keeps trying to make a score file somewhere that my normal account can't write to, and so it gives me an error about failing to do so every time I die (which is often). I know that the program can make score files, because a test play as root proved that. A quick check also showed that it doesn't save them in the directory of the program (/usr/games/) or in the /root directory, so I'm out of obvious places to look.

So I'm wondering if someone knows how I could track down where it's trying to save to and then possibly chmod the file so I could write to it as a normal user.

Probably in your own home directory. Look for hidden directories beginning with a dot (.) where programs usually store the settings for normal users.

It's not in my own home directory or the program would be able to write to it.

Ah, ok. I misunderstood your problem. Sorry. Did you check the /usr/share folder? Also check the /etc/ folder.

not in either of those places either.

I know this is minor necromancy but I had the same problem when trying to play this game. At
first it wouldn't save high scores or create save files, so I tried changing the permissions.
Then it would save games but wouldn't save high scores. In desperation I tried uninstalling it
and reinstalling it with one minor difference: I used sudo apt install bsdgames-nonfree instead of sudo apt-get install bsdgames-nonfree. That fixed the issue. Using Ubuntu 18.04.1.

I am not an expert, but if this helps at all then enjoy! I don't know why that worked, but it did.