One word: Overdubbing.
There are several alternatives:
1: Work out some harmony parts and record yourself singing them using a separate track for each part. If you run out of tracks, bounce a few down and overdub again on the freed up tracks.
2: A variation of #1: Record your friends singing the harmony parts.
3: You might also use outboard chorus FX to "thicken" the vocals at those points.
4: If you have extra money burning a hole in your wallet and want to experiment with another outboard FX box, Digitech has an outboard FX which is designed to synthesize harmony parts. However, I have not heard anything recorded with it, nor have I played with one myself to determine whether or not it is a worthwhile investment.
Actually, you only need three or four voices to get outstanding harmony parts. I do not know what genre of music you are doing, but listen to some barbershop quartet music or some southern gospel music (same harmonic principles as barbershop quartets but a different musical application.) You have one lead vocal and three harmony vocals. The blend works to provide a full, rich sound--with only four voices, no overdubs, and no techno-gimmicks.
Listen to some trios like the Lettermen or Crosby, Stills & Nash. I know the material is dated, but pay attention to what they're doing with the vocal arrangements. The CSN material is most interesting since all three singers are basically in the tenor range. No baritone, no bass. And yet the harmonies are still interesting.
