Actually, yes.
As pointed out by @ScottChamberlain, the reason you can't do it with the clipboard is that the program responsible for copy and paste in Windows is run in a user context. Until you log in, you have no user context, and it's not running.
However, that's not the only way to "copy and paste" text or access the clipboard. The password vault I use has a "paste into current field" option, and (to my surprise), it actually does pass this to the password field at the Remote Desktop login screen. Likewise, VNC and OoB-management technologies allow cutting and pasting into the login screen, because they don't use the unlogged-into RDP session's context to try to pass the clipboard contents.
Having said that, I don't find this feature particularly useful. Use a connection management program that lets you save your RDP sessions with username and passwords, and all you have to worry about is double clicking the connection.
I use mRemoteNG, which is freaking awesome. All my RDP, VNC, SSH, telnet, and even http/https connections are saved in there. Free and I'd pay hundreds of dollars for it, straight up. Got thousands of connections saved in there, and never have to type a password for any of them. (And before anyone says it, yes, my connections file is kept in a small Truecrypt container.) Yum, FOSS.
Best Answer
BGInfo from Sysinternals is a free program that's very common in enterprise environments. Configured to run at user logon, it can not only make the desktop look aesthetically unique, but can also display useful information about the server.
Also, you could consider pushing, via GPO, custom Powershell profiles and Cmd.exe settings that change their colors from the defaults on all production as well.
Just a couple ideas.
Edit: Beaten by a second. -_-