The exercise world is misguided in the belief that the only way to make an exercise harder is to lift heavier weight. Many practitioners fail to take into account the damage this does to your joints. If you continue to stress a joint in the same direction, you will impair joint function and lose the ability to reach your goal. (Learn how muscles get bigger.)
You can challenge muscles and joints in other ways by applying force from different directions. A lot of times, the weight load isn't as heavy, but it's applied more correctly. For example, consider the Bench Press. You perform the same linear pressing movement each time. Try performing a the pressing motion while kneeling or in a Lunge. This changes the muscle action and offers a new challenge to your muscles, but you are only using a fraction of the weight.
The "add more weight" mindset also causes serious technique issues. Often when people increase the resistance of an exercise, in order to accommodate the heavier weight, they move parts of their body that should remain stationary. They aren't strong enough to lift the weight with proper form, so the exercise does not yield the desired results and could lead to an injury.
To start applying force to your muscles from different directions, you must first learn how to control your body in a stable environment. Try doing a Machine Press or Bench Press and focus on performing perfect reps for a specified time. Maintain your form throughout the exercise, although it will become difficult as you fatigue. Try three to four sets of 30 seconds. Lower the weight slowly and do a hold in the middle of each rep to add muscle-building effect.
Next, I recommend performing exercises that force you to stabilize without the help of a bench or machine. Below you will see Dwight Howard performing a Physioball Single-Arm Press and a Seated Single-Arm Cable Press, both with lateral resistance. These will challenge your muscles in a new plane of motion and unlock strength you never knew you had. Start with three to four sets of 30 seconds each exercise.
Check out Dwight Howard's full workout.