Here is an intriguing challenge if you are looking to be able to perform an incredible feat. It is called the 100 Push-Up Challenge, and the goal behind it is to be able to do 100 push-ups in a row, without fail. I personally took up this challenge just for the fun of it, and to be able to say "I can do 100 in a row"! There are some really great programs out there that will help you achieve this goal, but this is what I have been doing, and I've seen my own results. It is relatively easy to start, but it takes some discipline to maintain (like any good workout should).

To start, simply see how many push-ups you can do, just do as many as you can, without resting. This is just to get your starting point, mine was 32. Take a minute to rest after this, and do a second set, try to match your first set, on my first try I only reached 25 on the second set, don't worry you'll get better. My goal at this point was to do as many sets to fail as it took to complete 100 push-ups. I finished in 4 sets with the last two having 25 and 18 respectively. I then repeated this as a part of my workout four nights a week.
 
Every time I repeated this portion of my exercise program my overall goals were to:
1)    Complete 100 push-ups in as few sets to fail as possible
2)    Increase the number of push-ups per set to need fewer and fewer sets, week over week.

Some people suggest that you set goals (for example to be able to do 50 push-ups in your first set by the fourth week), I merely decided to push myself to do as many as I could each time.
Now what I experienced was the interesting part. After the first week I started being able to do more push-ups in my second set than my first. I'm not sure why, but I consistently do at least 3 more in the second set than the first. I also discovered that by going to fail, I could go over 100 in 3 sets (40, 43, 25). Then there would be times when I would not be able to match the number of push-ups in my first set from my previous workout, but I would then turn around and surpass the amounts of my second and third sets. 
 
What I've learned is that there is no guaranteed rate of improvement, and everyone will experience different results. I am now in week 14 (and I haven't been consistent in doing the workout) and my 3 sets are at 50, 55, 50. This has been a plateau for me, and sometimes I can do up to 53 in my first set, but not on a regular basis. The make up for not doing better on the first set is the steady improvement of the third set! And I have viewed this as a sign that the first set will improve if I just stick with it.
 
I must also state that most professionals view this as a worthless challenge that offers no significant benefits. The workout only works a couple of muscle groups, and as a stand alone exercise program is unhealthy. My response to them: Duh! This is why it should simply be incorporated into the rest of your workout. This is a good way to get some growth to your chest, upper arms, and upper back, but obviously won't offer more than that. So if you want to have some bragging rights, consider adding the 100 Push-Up challenge to your exercise program.