There is, but it's a pretty high bar to clear. Merely exhausting one's appeals does not qualify as vexatious. He could keep going for years as long as his lawyers make vaguely plausible arguments each time.

Muddying the right of appeal with vexatious litigation is a very dark road to go down.

I'm sure people in the US are in no mood to take any advice from us here in the UK, having just recently abolished the right to a jury trial.

You are allowed your appeals. That's part of your constitutional right to due process.

There are some exceptions to this, but they are rare and usually only employed in extreme situations (aka things which would get the lawyers involved debarred).