In any case, Vim will display two rows of tabs when many files open immediately. I don’t want the arrows in the overflow style to display.
This is not exactly what you requested. But I switched from using tabs to plain old buffers with BufExplorer to switch between them.
http://www.vim.org/scripts/script.php?script_id=42
BufExplorer displays a good list of all buffer file names so you can quickly select one without the need for mousing.