Some companies did manage to produce NES compatible cartridges without NES permission, but it wasn't easy. They had to work around or defeat the lock out chip designed to prevent that.
Ironically, "Super 3D Noah's Ark", a bible inspired game, was one of such games, but for the SNES.
The required you to put a licensed catridge on top of the game catridge, and the unlicensed catridge would piggyback on the licensed catridge for loading.
