My lowest level possible challenge
Currently I am doing a challenge to beat FF1 DoS with the lowest levels possible, I will upload videos of boss fights to my youtube account: http://www.youtube.com/profile?user=aquatius and will only level if it is completely necessary, also I am running from all escapable battles.
Level 9 is the lowest possible
With 4 characters, level 9 is the lowest possible level to beat FF1 at. I know because I have done it. My only "cheat" was that I used a save state right before starting the battle with Chaos because it ended up taking me 25 fights to beat Chaos at such a paltry level.
The only experience you absolutely HAVE to get is by fighting the bosses and if you do the math you will find that that only gets you to level 9. However, without EXTREMELY careful planning you will meet enemies you can't run from and killing them will level you up to level 10 as Black Mage mentioned.
There is a method to avoiding these enemies. Final Fantasy picks a "random" number 1 through 8 every time you meet an enemy and it matches that number with a table that depends on which section of the overworld or dungeon floor you are in to determine EXACTLY what type of enemy you will meet. The trick is , these encounter numbers aren't actually random... there is a known sequence of 256 of these numbers that repeats over and over as you play the game. I came upon this 256-number sequence in a walkthrough and using it (and a lot of trying different routes through dungeons) was able to plan out a way to walk through the entire game such that I would never have to fight any battles that I couldn't run from. Very difficult, but possible. It took a lot of work but, considering this is my favorite video game of all time, it was well worth the effort.
If any of you would like to attempt it, I will happily provide further advice. YouTube didn't exist when I completed this challenge, but now that it does maybe I will make a video showing my characters beating Chaos at level 9.