After you are rescued from your initial encounter with the Man on Fire in the prologue, you must follow someone down a corridor. There's a rail to your left, which lets you pull yourself from the floor. Look closely at the cork board on the wall to see a poster referencing Kojima Productions and inviting you to "Join the Fox Team." There's also a booth number: CP 2308. That booth that was assigned to Kojima Productions during GDC 2013.
Paz Lives
After the Medical Strut has been constructed, make your way to the top and you'll spy a door with a blue light. Inside that room, you'll find a "Work in Progress" sign to your left. Head right instead, where you can enter another door to find Paz. From there, it's possible to have additional interactions and uncover new information as the game progresses.
Rescue Hideo Kojima
You can rescue Hideo Kojima, the creator of Metal Gear (and this game's director), in the Side Op 112 Intel Agent Extraction mission. Once you have done so, he'll head back to the Mother Base.
Silent Hills in Africa
If you played PT, the demo that was released to promote Kojima's now-canceled Silent Hills project with Guillermo del Toro, you likely heard a recording that outlines a horrific murder. In Metal Gear Solid V: The Phantom Pain, you can find and listen to that same recording. Look for a radio positioned on the floor near one corner of a tent, just beyond an outpost encampment in Africa (it's near the site of Mission 20: Voices).
Smelly Snake
Complete enough missions without returning to Mother Base, and you'll eventually see flies swarming around Snake. When you finally do head back to Mother Base, people will greet you if you get close but also will try to avoid you. When you talk to Ocelot, he'll say "Snake, you stink" and will dump a bucket of water on you.
The Cake Is For You
Kojima and team apparently decided it made sense to reward players by letting them have cake. When you start a new game, you'll be prompted to enter your birthday. If you play the game on your birthday, there will be a party waiting for you at the Mother Base, which includes a birthday cake and cigars.
Kojima Productions Lives...