The push-ups partI have done push-ups on a regular basis for most of life, since I was an adolescent. They work out a lot of muscle groups at once, and don't require much time, space, or any equipment. My limit has always been "about 60". That's how many I could do in a row when I was at my peak of fitness in college, that's how many I could do in high school, that's how many I can do now. It is interesting (and somewhat frustrating) to me that the number never goes up. To be fair, in college I think I could hit 67 in a good set, while now my top number is 64. Still, not much difference.
There is a website called One Hundred Pushups that lays out a plan to work up to doing 100 push-ups in a row. According to their schedule, if you can do 60 push-ups, you're on week 6, the final week. You will do multiple sets of push-ups, with prescribed rest periods in-between. No doubt this is the key to building up endurance. However, when I tried, I was unable to even come close to doing the prescribed workout, which is (again) instresting and frustrating.
Anyway, five sets is too time-consuming, so I've been doing three sets: as many as I can do (about 60), 30 seconds rest, as many as I can do (about 13), 30 seconds rest, as many as I can do (about 10). We'll see if it leads to increasing my maximum. I know it has to be possible. Look at all the records!
Another exercise I like is Burpees. Again, it works a lot of muscles at once, in a short time, and requires little space and no equipment.