What happens if I take two strong magnets and press their like-poles towards each other - does the repelling force created by them equals the force I'm exerting on them or does the repulsion force create altogether a stronger force? I'd appreciate a simple answer as my knowledge of mathematical formulas is quite poor. Thanks! Ami
The closer together you push them, the stronger the repulsive force between then and the harder you have to push - or the more force you have to put on them to push them together.
Strictly speacking if the force of you exert on them is exactly equal to the repulsive force between them (and your force is in the opposite direction exactly) then the magnets will not move because the total force on them is zero. So to push them together and make them move close together the force you exert have to be a bit larger than the force fo repylsion between them.
As the magnets get closer and closer together the repulsive force gets bigger and bigger.
Eventually if the magnetcs actually touch each other there will be the magnetic repulsive force and the force due to the contact between them - like if you try to push two blocks of metal together.
In answer to your qusetion - the more you push them together, the closer they will get and the stronger the repulsion between them. The repulsive force just keeps rising smoothly as they get pushed together.
If you push magnets closer to each other then by newtons third law your force should be equal to magnetic force... provided system is at rest. if you go closer magnetic field does increase so does the magnetic force.
The formula to calculate the force between two magnets includes separation distance in the denominator. When the separation distance becomes $0$, the force is infinity. Even if a small separation distance is inserted, the force is determined to be very high. Two 3 cm neodymium (#52) magnets, same poles forced together, can reach forces of 1 to 2 million newtons.