What is minimal? We can minimize equipment, we can minimize time, but to determine the minimal exercises we need to know more about who is asking the question. Someone who is morbidly obese with emphysema would have a minimum workout of "walk to the end of the driveway every day". An Olympic athlete might have ten workouts a week just to keep from regressing. My point: fitness is relative to a specific task for a specific person, so minimal needs vary widely.
So, here are some variations on the theme of minimalism.
The Luddite Workout
When we talk about minimal exercise programming, the first thing to go is all the equipment. It may not be optimal to eschew dumbbells, barbells, squat racks, bumper plates, kettlebells, benches, rings, pull-up bars and everything else, but it's certainly possible. Most dedicated people can get quite fit and strong with nothing but their own body. So out the window the equipment goes, if we're looking for "minimal". We assume only the use of your own body and the world around you.
So what are the most productive exercises to do with just one's nude body and gravity? Well, I wouldn't omit running and sprinting, push-up progressions (from on-the-knees negatives all the way to explosive triple-clapping push-ups and one-armed push-ups), pull-ups and chin-ups if you can find a tree branch or convenient ledge, handstands and flips and other gymnastics, squat progressions (from holding onto a pole to do a third-world squat all the way to jumping single-leg squats for height and distance) and dips if you see fit to allow a pair of sturdy chairs into your arsenal.
Of that list, I would say that there is no "must-have" exercise. If you want to do all upper-body strength work and leave out all the running and squatting, that could work. If you want to just run, that will work too. But optimally I would do at least the following:
- something fast that makes you winded, like sprinting or picking up or carrying random heavy things (e.g. tires or logs)
- something to make your legs strong, like single-leg (pistol) squats and jumping
- some upper-body pulling, like chin-ups
- some upper-body pushing, like handstand push-ups or dips
Notice that most lists of bodyweight exercises very quickly start using at least some equipment. But technically you don't need a pull-up bar or chairs for dips; it just really helps.
Bodyweight Workout Resources: I am partial to Coach Sommer's Building the Gymnastic Body book (here's a summary article), though I've heard that the prolific and inventive Ross Enamait's Never Gymless is good, and that Convict Conditioning is pretty good (book, summary image).
Best Return On Investment
For the best results in a minimum amount of time, barbells can't be beat, particularly for strength. A user on MetaFilter distills strength training down to this minimal program:
A: Squat 3x5, (Overhead) Press 3x5
B: Deadlift 1x5, Bench (Press) 3x5
That's probably the simplest, most minimal full-body routine you can do and still cover all the bases. You should be able to do it in 30 minutes or less. Do it 3 times a week, alternating A and B each workout. Do a few lighter warmup sets first, then do all your work sets at the same weight. Rest 3-5 minutes between work sets. Start with a weight you know you can accomplish and add a little bit every time.
As described, it requires a barbell, some plates, a bench, and preferably a squat rack. (Technically, one could clean the weight to a front squat, but that requires at least one additional skill.) Personally I would substitute pull-ups or dips for bench press, which would allow the use of more readily-accessible chairs or pull-up bar instead of a weight bench, and remove the danger of doing bench presses without a spotter.
I'd say that such a program would be plenty for a basic level of strength. To get to fitness, I'd add a little running (say, half a mile or less, as fast as possible) or other cardio workout, and address any mobility or flexibility issues that arise from the workout program as described. That's it. Strength, conditioning, mobility.
Barbell Training Resources: the best introduction to barbell strength training I've yet read is Rippetoe & Kilgore's Starting Strength (book, unaffiliated and mostly-correct wiki summary).
Minimal Equipment (But Not Bodyweight-Only)
If getting a barbell and squat rack seems too costly, but you want the rapid return-on-investment that external loads provide for resistance exercise, I'd buy one kettlebell, as heavy as you can safely handle. This should be between twenty-five and fifty pounds for untrained adult men and women. That plus a pull-up bar or gymnastic rings allows for tremendous external resistance (though not as much as with barbells).
With this kettlebell-plus-pull-up-apparatus setup, one could do pull-ups, goblet squats, kettlebell-style swings, clean, and snatches, as well as overhead presses, Turkish get-ups, and with some practice, single-leg (pistol) squats, eventually with the kettlebell. That's a heck of a lot of strength and conditioning potential.
The bare essentials of that list would be some form of squat (goblet or pistol), plus some form of press (overhead, or a Turkish get-up), plus either swings, cleans, or snatches. They can be combined with rests for maximal strength or as a complex for strength and conditioning.