River City Ransom has all the elements of a great game you still find in modern gaming: an immersive storyline, character development, leveling, and progression, an inventory system, and an ability to purchase upgrades. It was one of the first open-world action games, and has had an incredible legacy on other consoles, which makes its absence from the NES Classic Edition a true head-scratcher.
On the surface, it may look like any other side-scrolling beat-em-up, but it's far more interesting than that. The player takes control of one of two brothers on a mission to save the girl who has been taken by a rival gang all the way to the other side of River City. While it was somewhat limited, due to the hardware at the time, the story was nonlinear, which allowed the player to explore the open world in a sandbox-style of gameplay. You could venture back to levels you had already beaten, and hold off on progression to level up for as long as you wanted.
Since the game was such a success in the Japanese and Western markets, it has since been ported to just about every other console since. Most recently, a port was released on the Nintendo 3DS, and has done very well, but for now, if you want to play it on the NES, you have to take your chances with a dusty old cartridge at your local used game shop.