I am looking for a low cost mp3 player, or some other digital player,
that I can use to play a message in a loop for my on-hold messages. 
My phone system has a plug for the input which is exactly like a
headphone jack, so most players are okay on this point.  I'm having
trouble finding one that runs constantly that does not run on
batteries.  I need one that plugs into a wall, so it can run
constantly. Obviously, I would like the cheapest alternatives
available.  Thanks in advance.

ewek1
I have seen MP3 players which have a USB connection, thus I guess you
can power it via a PC if you have a PC near your phone system. I am in
the UK, so I found a description on my favourite audio component
website (richersounds), you can see the specific page at:
http://www.richersounds.com/index.php?f=itemdetl.php&p=300826
It looks like there is a verison 64 & 128 - presumably these refer to
Mb storage. I note that both these players are out of stock, but maybe
you can find something similar if the USB power is an option??

Perhaps you can plug the MP3 player into a self-powered USB hub,
thereby eliminating the expensive need to include a computer into the
system.