Let's see how many games we can list that utilize this trope.
Streets of Rage 2: 1-Up hidden behind the foreground layer in Stage 1
Doom 2: When you start, just turn around and you'll get a chainsaw
Duke Nukem 3D: Rocket Launcher hidden near a billboard at the start of Episode 1, Devastator hidden in a wall at the start of Episode 3
Morrowind: In the census office during character creation you can take a gold plate which is worth a lot of money. No idea if this was an intentional design choice, but I do it all the time to start off with some quick extra cash.