I accidentally left my game on over a week and now my save file is up over 180+ hours. I really dont want to start over. I am a level 13 and I am worried that if I keep playing the file will become unstable. Any suggestions on whether or not I should scrap it and start over?

Yes, start again after every patch. And if the game shows even a single hint of unstable, start again. Afterall, it's not Bethesda's fault they have shipped out broken game. It's our consumers fault

Bethesda needs to patch the software to limit the number of saves to say 10 then start reusing the oldest. If you look at the game save files, under Saved Data Utility it gets huge unless you go in and delete the older files yourself. This is time consuming since you can't remove multiple files just single file one at a time.

Holy crap! Forget the game! I'm surprised your console isn't fried!

Jay, I signed on just to give you a straight answer. My game saves are over 400 hours, I forget because I haven't really played in a month. Of course I didn't get a couple hundred hrs at one time like you did, but I played Skyrim on just 1 character, created 11/11/11. I think you'll be ok but level 13 is pretty low & you can easily restart & get back to there.

leaving the charachter on shouldnt really have done much to be fair just not very good for your PS3
the time left on and stood still shouldnt do anything to your game or save ( i say shouldnt but remember the guy with the 80+MB save file?) its just a time stamp my 11/11/11 character has 260 iv given up now though due to glitches and issues caused by me doing everything possibl on one save (still the last couple of bits to do and its 100%)
whats the file size? did it go up a stupid amount or down even?

also play it and see how that file is thats the only way youll know the week of stood doing nothing has effected it or not

As has been said, the simple fact the time played has clocked on as if you had been playing all week will have no effect on the stability by itself. The overall save file size is really only a rough indication of how much of the game world has been explored and what items found / stored / dropped and not flushed.

A console left on all week will be fine, it's almost in an idle state with nothing changing on screen so the built in temperature control will keep the console temp within it's operating range quite happily, especially if it's a slim. Assuming no cooling failure scenarios of course but then I don't know if the console has a fail safe thermistor trip cct.

The file size stayed pretty much unchanged, at around 5MB. Hopefully my ps3 is ok. Its the slim and it looked like it had gone into some form of hibernated state. However, the game was still clocking for some reason.

Probably has a suspend mode for just this situation, well found.

In fact if you have your controllers set to power off after a time, once that happened, the game went to background. Bit odd that time still passed though as it's normally a suspend for games.

Good luck playing on and avoiding the bugs.

As long as the file didn't bloat up then you should be fine... If it does bloat up to some huge size then it is a good idea to restart but otherwise you should be able to simply continue and not worry about it...