Why can't I bleed my brakes properly?: 560SEL
 

I tried with the Motive power bleeder and bled each caliper until no bubbles were seen. The brake pedal is still mushy after many attempts.

I tried it with the engine on, engine off, just ignition on, pumped the pedal and left the pedal alone. I started with the rear passenger side, then rear diver side, then front passenger side and finally the driver side. I tried at different pressures, 15 psi all the way up to 30 psi.

What is it that I am missing?

PUSHING or SUCKING?
Make sure you have a power bleeder, that is necessary to force enough fluid through the ABS system. If you look under the hood at your ABS unit you will see the brake lines come out the top of the unit, that is where the air gets trapped, you can pump till the cows come home and never move enough fluid to force the air to go down and out the bleeders.
Once it is done correctly you will have a rock hard pedal and a smile.

I did it all using a power bleeder. It's simply not doing for me.

Just for clarity - you are forcing fluid through the brake reservoir not sucking out of the caliper nipples.

Yes

Are you losing fluid? Maybe a rusted brake line?

Nope, no losses anywhere. I have driven the car as is for last two weeks and the reservoir is still at the same level.

Just another thought - Look at the rear "soft lines" and make sure either line is not swelling when you apply brakes. I have had that trip me up before.

Both are brand new! The reason I am doing this this time is because I replaced on of the rear calipers. I have flushed and power bled the system before and had it workingjust fine but this time I am puzzled.

You are aware that the reservoir has two compartments ???

There is only one inlet to fill the reservoir and interface with the power bleeder. What is it I am missing?

One cap, but there is a wall in the res that seperates the front from the rear
systems...in other words , there are two seperate reservoirs.
So, what happens is sometimes the level is not high enough to get over that wall/seperator and you wind up with one res pushing air down the line while the other is fine.
The procedure is to make sure both containrs are completely full before bleeding.

The reservoir was full to the brim. There were no air bubbles coming out of the rear ones. New brake fluid was flushing out at the rear without any issues.

What prompted you to bleed the brakes?

New fluid coming out the rear can still be air entering at the reservoir as you bleed...That was my point.
You have to make sure the fluid level stays high in the res as you open the bleeders..and then recheck the level before going to the next bleeder.

The other possible is you have an internally leaking MS [ fluid bypassing the piston cups under pressure.] You can usually tell that by holding the pedal down hard for a min. and watch for sinking pedal..........