My bet is that the master cylinder is shot (leaking around the piston). If you have really bled the brakes properly, AND there isn't an obvious fluid leak, I'd really suspect the master cylinder. It could be a problem with one of the calipers, but that would be more unusual, and repeated pumping of the brake pedal would temporarily "fix it" and firm up the pedal - if it doesn't firm up, my money's on the master cylinder. Even with a bad leak on your master cylinder, it'll push plenty of fluid out to your brake calipers for bleeding (since there's no real pressure on the system at that point).
OTOH, if - for example - you had run the brake fluid way down to the point you got lots of air in the lines, AND you somehow didn't get it entirely pushed out during the bleeding process, you could be chasing your tail (and a new master cylinder won't help). Just throwing that out FWIW, since I have no idea of the OP's mechanical aptitude.