I was playing my friend's FF deck and this came up:

Turn 4, he recruits "Alicia Masters" on his initiative. He also has 4 other guys on the field(A one drop, 2 drop, and 2 3-drops.) Needless to say, His Alicia Masters was kept alive.

Turn 5 comes about. I recruit for the turn, and pass priority, starting his build step.

For his first move of the turn(after the resource was laid) he exhausts his Alicia Masters for her effect.(Torch and Thing cost 1 less to recruit this turn)

Next, since I pass, He plays Cosmic Radiation from the resource Row, and again, Uses Alicia Masters' effect.

Next, he recruits a 7-drop Thing. Is this legal? I'm not sure if Alicia's effect only works once.

It's legal. Doesn't say "use only once per turn" on the card.

I team them up with Heroes United and use X's School for Gifted Youngsters to ready Alicia.

Alicia Master's effect is cumulative, so if you are able to ready her 3 times, during turn 3, you can recruit Thing 3 drop absolutely free.