There are a few exercises you can do but don't expect huge gains from them
1) Push ups. You can vary this exercise into other more challenging versions such as 1 armed push ups, diamond ups, do push ups between two chairs for a greater range of motion, etc.
2) Lift gallons of water. They can act as dumbbells. Drink to reduce weight lol
3) 1 legged squats. Start with two legged squats to get a hang of it.
4) Cardio. Get ahead on your cardiovascular system. Play sports, run, cycle, swim, none of those require a gym membership.
Just to name a few. However, besides cardio, these exercises won't get you massive gains. You could just do cardio and dive into your diet while waiting to purchase your equipment.