who is the only guy wearing a hat on the back of the two dollar bill?

Answer 1

It was actually John Dickenson, although it was believed for years to be Stephen Hopkins.

~~We assume you're referring to the 1976 version of the $2 bill that features John Trumbull's "Declaration of Independence" painting. Created between 1788 and 1795, the painting memorializes the document's historic signing. However the scene didn't actually take place as depicted. Trumbull included all the key players in his painting, whether or not they supported independence, and whether or not they actually signed.
Twenty-five years after he finished the painting, Trumbull identified the man in the Quaker hat as Stephen Hopkins, a signatory of the Declaration of Independence and first governor of Rhode Island. But in the 1940s, Trumbull's original sketch of Hopkins was discovered. It called into question the identity of the man in the wide-brimmed hat.

**Art historian Irma Jaffe concludes the mysterious man was actually John Dickinson, a Pennsylvania Quaker opposed to independence. A commission from the Rhode Island Governor's Office investigated the claim and agreed with Jaffe's findings. **

Maybe John Trumbull intended to be vague. Or maybe he was pulling a joke. He certainly showed a sense of humor when he painted John Adams stepping on Thomas Jefferson's foot.

Answer 2

It's Stephen Hopkins.