I really like the question as its something that a lot of teenagers struggle with at this age.
To your question "Will increasing the number of reps I can do back to back help with muscle gain, or will this gain plateau eventually?" the answer is Yes you will gain a drop of muscle but not alot, and Yes doing the same thing all the time will cause you to gain a plateau.
Having said that I think you need to look at other options ifd you are serious about getting in better shape, the best option for you would be is to look to get a program, and I will list 2.
would be to get something like INSANITY or T25 if you are looking for something less intense, and thew are both from Shaun T, the benefits of these programs are, one that they do not use any weights, just a band occasionally, and 2nd and most importantly using a program will help you better keep track of your progress and make sure that you don't hit a plateau, but again you mentioned that Buying weights is not an option and if its because of financial reasons this would probably not be an option as well.
would be to follow a guy who I have followed his name is Matt Furey, it will be very cheap and it will help you gain overall strength, some muscle as well.
Here is a link to his website http://www.mattfurey.com/index.htm
You can also read the following to get more insight http://www.bodybuilding.com/fun/mahler19.htm
Routine #1: Killer 200 Pushup Workout: 3 minute back bridge 100 Hindu Squats 50 Hindu Pushups German 50: 10 Elbows in, 10 hand over hand, 10 Superman, 10 hands wide, 10 hands backwards performed in a row, without stopping. 25 Hindu Pushups with feet 10" apart 25 Hindu Pushups with feet together 25 Hindu Pushups with hands and feet together 25 fingertip Hindu Pushups 3 minute back bridge, 3 minute front bridge.
Routine #2: 3 Minute Back Bridge 100 Hindu Squats 10 Grasshoppers 35 Bootstrappers 10 Kneeling Back Bends 20 Hindu Jumper Squats 10 Mountain Climbers.
This is officially a beginners routine, but for most this will be too much, so do the sets as listed but with less reps.
I hope all this helps you, I will list a few tips that will help you along.
I don't have your lean factor but from the info you provided i would say that your calorie intake should be at least 2000 calories a day not less.
Don't rush into anything as it will do you more harm then good, try to find someone who has done these programs and get more insight as to if the intensity level it will suit you.