Will President Trump seek to have his likeness added to Mount Rushmore? Will he announce the existence of alien life?

Whimsical as these situations might sound, there is a growing market for betting on an already unpredictable Trump presidency.

Paddy Power, the Irish gambling website known for its over-the-top marketing stunts, says wagers associated with Mr. Trump have been more popular than any other novelty bets it has offered in the last year, including bets associated with Britain’s referendum on whether to leave the European Union.

Now, Paddy Power is hiring a “head of Trump betting” to oversee bets related to the American president and his administration.