My Gym Bag

I am very fortunate to lifts in a fairly good sports facility here my college. We have a Glute Ham Raise, plenty of platforms and racks, and loads of weights and dumbbells. There are 3 essential things that I bring along that are critical to my training that are not provided and I can't do with out.

Bands
I personally think that these are one of the most versatile tools ever. With all the variety of resistances you can get ones for dynamic work, activation, prehab, and they come heavy enough that you can use them for an intense workout. You can also attach the minis around your back and add accommodating resistance with DB work.
Sample Band Movements
Act-Lateral Band walk
Act-Standing hip Ext Rotation
LB-Band Good Morning
LB- Band Pull Through
UB PUll- Bent over Rows
UB Push-Push ups

Heart Rate Monitor/Stop Watch
This device I utilize for my own training in regards to whether I am over anaerobic threshold or not. This is also a great gauge to see how you are recovery between bouts of exercise. For example: Track the peak heart rate following an exercise, lets say the squat. If you do 3 reps and HR is at 140 and then in the following weeks your 3 reps peak HR is above or blow that you can track whether or not your work capacity is increasing. The stop watch is to keep honest with rest intervals and keeping track of how long certain bouts of exercise take.

Nike Frees
This is my personal shoe of choice because they are so dang light and comfortable. I also like the "bare-foot" feeling you get from them. Personally it doesn't really matter the shoe as long as the heel isn't to big. Also the more feeling you have in relation of foot to ground contact, you seem to get better proprioception and awareness of foot position.

Things you may want to consider that you might not have provided: Foam Roller, Tennis ball, Tiger tail, Lifting Straps, Airex pad, Boards (for board press), TRX.