WASHINGTON — Since Bill Clinton occupied the White House, the commemorative medallions known as challenge coins have been stately symbols of the presidency coveted by the military, law enforcement personnel and a small circle of collectors.

Then came Donald J. Trump.

His presidency has yielded more — and more elaborate — coins that are shinier, flashier and even bigger, setting off a boom for coin manufacturers, counterfeiters and collectors, with one official Trump challenge coin recently fetching $1,000 on eBay.

Among those produced in recent months by members of a White House military unit is a coin featuring Mr. Trump’s private Florida club, Mar-a-Lago, on the front, and the presidential seal, the White House and Air Force One on the back. Another has Pope Francis on one side and the president’s face set against the White House on the other.

And Mr. Trump’s aides have commissioned multiple versions of an official challenge coin , for which the president himself has reviewed several proposals, according to people familiar with the process.