I want to be able to remotely access the desktop of my Raspberry Pi 2, running Rasbian Full, from outside of my LAN network. I've been able to easily connect on my network, but I want to be able to access it when I'm not at home too. I am not able to port-forward or doing anything else requiring access to the router options or settings. I've seen some tutorials online but the ones I've found I seem to miss steps from place to place.
You are asking something like "I have a door with a lock but no key". Breaking the lock (using an exploit) I don't see as valid solution. I see mainly two solutions: purchase your own controllable router with internet access or use a reverse tunnel.
For direct connecting to the RasPi you need port forwarding. This can be done with your own router you can manage.
If this isn't an option you can use an encrypted ssh reverse tunnel. For this you need a third device on the internet with a static public ip address that you can connect from the internal RasPi. You can use a second RasPi or much easier you rent a very cheap virtual server on a provider (that's what I have done). Now you can permanently establish the reverse tunnel from the internal RasPi to the external third device. With its known public ip address you can connect from everywhere on the world to it. The reverse tunnel will redirect all traffic to the internal RasPi.