BuddyPress.org

Here’s how I do it:

Print out the Username:

echo bp_get_loggedin_user_fullname()

There is also a related function that avoids the need to ‘echo’ that function. Search your bp files for this function and it’s immediately above or below that one.

Print out a URL to their Profile page:

echo bp_core_get_userlink( bp_loggedin_user_id() )

Although I think that gives you a hyperlinked userlink – if you review that function you should be able to figure out a way to output the raw url, if that’s what you are looking for.