Windows Server 2008 R2 RDP login paste password
Solution 1:
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.
Solution 2:
Contrary to what the other "answers" for this question say (including the accepted one), there is indeed an easy way to do this, also if you (like me) for different reasons (most likely security related) don't want to save the password in the RDP connection shortcut/file on your computer.
In the RDP connection dialog (i.e. pre-connect), enter the username, and also put the checkmark in the option "Allow me to save credentials" (don't stop reading here, we will NOT actually save the credentials after all!).
Now when you press connect, you will be presented with a system-local credentials dialog, where the username is pre-filled, the password box is empty but possible to paste right into without any fuss at all, and the option "Remember my password" is pre-checked.
Now, simply paste your password, and also UNCHECK the checkbox for "Remember my password", then click the "Connect" button.
Bam, you're in, and the password will NOT be saved locally after all!
Please accept this as the real answer for this question, instead of the currently selected non-answer.
Solution 3:
Install and use a newer RDP client. Windows 2008 supports NLA. With NLA you can provide your credentials to the RDP client before the RDP session starts. Copy and paste works fine in that case.
Or you could even just save an RDP file per host with stored credentials.