Is it possible for a Mob to spawn *right next to* the player in Minecraft?
Unless other changes have been made to the spawning code, enemy mobs can't spawn within 24 blocks of your current location. The exception to this is of course mob spawners, but that's a different story.
They can't spawn right next to you; you have a 24-block radius of safety. They can approach you afterwards, but that's different.