But that didn't stop the Trump administration from reportedly asking.
Last year, a White House aide contacted the office of South Dakota's governor, Republican Kristi Noem, to inquire about the process for adding more faces, according to The New York Times. The Times cited a Republican official familiar with the conversation.
Trump's interest in Mount Rushmore, and his desire to be etched in among the four existing presidents, is no surprise to Noem, who says he brought it up during their first meeting in the oval office.
"He said, 'Kristi, come on over here. Shake my hand,'" Noem recounted to South Dakota's Argus Leader. "I shook his hand, and I said, 'Mr. President, you should come to South Dakota sometime. We have Mount Rushmore.' And he goes, 'Do you know it's my dream to have my face on Mount Rushmore?' "I started laughing," she said. "He wasn't laughing, so he was totally serious."
Noem, a close ally of President Trump's, has since played into that dream, at least with the power she has.
When Trump visited the monument in early July for an Independence Day speech, she greeted him with a four-foot model of Mount Rushmore with his face carved into it, the Times reported, citing a source familiar with the exchange.
The news came amid rumors that Noem could overtake Vice President Mike Pence as Trump's running mate in the 2020 election, though The Times cited a source who said Noem has already indicated to Pence that she isn't trying to replace him.
To be clear, adding another face to Mount Rushmore is not possible. While it looks like there's a spot to the right of George Washington — which is actually where the sculptor, Gutzon Borglum, intended to put Thomas Jefferson — the rock surface is unstable.
Noem reportedly joked with Trump that there is another option, Argus Leader reports.