The moment where Akeem and Lisa truly connect for the first time. After Akeem offers to pay for their first dinner out together, Lisa refuses his offer in a truly touching manner:
Lisa: Akeem, I wish you'd stop worrying about being poor. If I wanted a wealthy guy, I'd be with Darryl... (she places her hands on his) ...not you.
Cue Akeem and Lisa's dance to the tune of Jackie Wilson's "To Be Loved", their First Kiss and Akeem's truly dreadful rendition of said song in the streets.
After Akeem tries and fails to persuade Lisa on the train, after Lisa leaves, an old woman walks up to Akeem and tells him "If you're really a prince, I'd marry you.". Still upset, Akeem humbly gives her the earrings Lisa throw at him in gratitude.
Another one that works as a Crowning Moment Of Awesome is the scene where Lisa's father threatens to break his foot off in King Jaffe Joffer's ass. Made more touching considering that throughout the whole movie, he's been trying to get her to marry a rich douche, but when offered two million dollars instead of an apology from the king for breaking his daughter's heart, he rips the check up and tells him to quit insulting his daughter.
Akeem would willingly renounce his royal birthright and all the privileges it brings — and there are rather a lot — to make Lisa happy. She decides she'll put up with being a beloved and ridiculously wealthy princess.