This is an idea for a 2D sidescreen adventure video game. You play as a guy who hits his head and gets amnesia, kleptomenia, and more importantly compulsive hoarding. It is a comedy puzzle game where you must handle everything the world throws at you not metaphorically but physically.
In the openening scene you see the unnamed main character walk through a backalley and slips on old greasy fish & chips paper, he barely safes his fall and counts his luck. As he walks on a flower pot falls on his head, giving the player an insight in the double-edged sword of luck.
John Doe wakes up without realizing what happened in an alley and miraculously has no blood loss. He says to no one in particular "I really should've brought along some painkillers, I've got a splitting headache.". He walks one step and continues "I'm also quite hungry, why do I keep forgetting to bring along food. Next time I won't leave anything behind!". From here on the game begins kinda like a point-and-click adventure, with an introduction scene of a shopping street after the alley. You can pickup stuff but you can never remove it from your inventory. If you stand near something too long John will automatically pick it up or steal something from people around him. This will be indicated with a small progress bar on the item or on the person he's going to steal from.
Afterwards John will slowly realize he has no idea who he is and where he is from. This will be the main goal of the game, to find out where you come from and who you are. Slowly you'll find out some very strange and funny things about him and where he is from.
Not being able to drop items will end up weighing you down and you'll have to manage his inventory space. Each item has weight and size and you'll have a square-based space in which you can lay them out. Some small items can be stacked. The only way to lose items is by using them to construct other items, or use them up. The more eleborate and complex the solution, the better, because you'll lose more items that way. He will keep broken items because you'll never know when you might need it, these are very hard to get rid off.
You can carry more by putting items in e.g. a shopping bag, purse, briefcase, backpack, bag-of-holding, or box. And you can place those in/on a cart, another person, bicycle, car, mule, or other mobile carrying platforms to keep them near you. Some inventory spaces are thus considered near-you which is acceptable to his hoarding compulsion.
If you carry too much you won't be able to move. You'll have to solve this through your inventory. If you can't solve it and you can't move you'll simply die on the spot. Savegames are thus a must ;-)