I record a lot of radio programs off the internet, usually in one or
two-hour lengths. I would like to transfer them to a portable MP3
player that would allow me to do the following:
1) play back a one or two hour program that I have recorded.
2) mark the beginning and end of selections within that program.  For
example, if there are 15 songs in the hour and I only want to save the
3rd, 9th, and 14th song.
3) back at the desktop computer, save those marked selections and
delete the rest of the program. (or even do this right on the portable
unit to free up storage space instantly, if that's an option.)
If this feature even exists on MP3 players, I'm seeking the answer to
this question: what are the best of the MP3 players on the market that
offer this feature.

I don't believe that there is an mp3 player that will let you do this
kind of editing on the mp3 player itself.  I would recommend that you
open the recordings on your computer using audacity, a free audio
editor available here: http://audacity.sourceforge.net/.  You can use
that program to split the audio file into chunks containing the songs
that you want.  You can then delete the files containing the songs
that you don't want, and transfer the ones you want to keep to the mp3
player.

Thanks for the tip. I'll definitely check it out.
Just to clarify: Although editing on the portable would be nice,
that's not the function I really need. What I really need would just
be a way to bookmark selections on the portable. The editing can be
done back at the desktop.
What I do now is write down the beginning and ending times of the
portions I want to select out, and then hunt for them by time, back at
the desktop.  It seems ridiculous that such an awkward, clunky,
time-consuming (and dangerous, since I do this a lot in the
car)procedure should be necessary in this age of electronic devices
vying to top each other's feature sets at an exponential pace.