Is there a "vim runtime log"?

Put this function into .vimrc:

function! ToggleVerbose()
    if !&verbose
        set verbosefile=~/.log/vim/verbose.log
        set verbose=15
    else
        set verbose=0
        set verbosefile=
    endif
endfunction

Then create directory ~/.log/vim and call ToggleVerbose() to get your log in ~/.log/vim/verbose.log. Note that you may catch «variable nested too deep for displaying» error which will not normally appear just because you have raised your verbose level.

I don't think there is a runtime log, per se, but you can run it in debug mode.
http://web.archive.org/web/20090323034339/http://www.troubleshootingwiki.org/Debugging_Vim_Scripts

:messages shows all warnings, errors, and informational messages that appeared (possibly briefly) in the vim statusline.

:echo errmsg prints the most recent error message.

g< is another feature few people know about. From :help g<:

The g< command can be used to see the last page of previous command output. This is especially useful if you accidentally typed <Space> at the hit-enter prompt.

For example try :!ls then cancel the prompt, then hit g<.

running vim with the -V[N] option will do a pretty hefty runtime log, here N is the debug level.

vim -V9myVim.log

would create a log of debug level 9 in the current directory with the filename myVim.log