How to allow the reply link to remain on the comment form after I have reached my 10 nested comment limit?

If you need to modify the max depth, you could use the thread_comments_depth_max filter:

/**
 * Set max comments depth to 15 on the discussion settings page
 */
add_filter( 'thread_comments_depth_max', function( $max )
{
    return 15;
} );

then the dropdown on the discussion settings page will show the range 1 – 15.

But I can imagine very deep comment threads would be harder to read.

Another approach would be to override the thread_comments_depth option value on the front-end:

/**
 * Set max comments depth to 15 on the front-end
 */
add_filter( 'option_thread_comments_depth', function( $val )
{
    if( ! is_admin() )
        $val = 15;

    return $val;
} );

The reason why the reply links don’t show for depths greater than the max depth is this part:

if ( 0 == $args['depth'] || $args['max_depth'] <= $args['depth'] ) {
    return;
}

within the get_comment_reply_link() function.

It’s not possible to use the comment_reply_link_args filter, because it’s applied after the depth check, for some reason.