What legal problems might you run into?
Well, you'd be violating 47 USC 301, which requires a license for anyone broadcasting in the United States. Penalties for that are given in 47 USC 501 (a fine of up to $10,000 and up to a year in prison), 47 USC 502 (an additional fine of $500 per day of violation), 47 USC 501(b)(2)(D) (forfeiture of up to $10,000 per day, with a maximum of $75,000), and 47 USC 510 (forfeiture of your equipment).
"Just outside the FM band" on the low end (below 88 MHz) is television channel 6, while on the high end (above 108 MHz) are frequencies used for airplane navigation. If your transmission interferes with a licensed user of channel 6, you're also in violation of 47 USC 333, which carries much the same penalties as violating section 301, but without the forfeiture of equipment. If your transmission interferes with air navigation, you're in violation of 49 USC 46308, which carries a penalty of up to five years in prison and a fine of up to $5,000.
Honestly, you'd be better off broadcasting inside the FM band: there, you can broadcast without a license under 47 CFR 15, which limits you to a broadcast power of 250 microvolts per meter at 3 meters (basically, you can broadcast to anyone in your house or maybe the next house over).