Is it even possible to have such kind of a collision?
Yes, when you fire a pistol the hammer hits the bullet with a relatively small initial kinetic energy but the kinetic energy of the hammer and bullet after the collision is considerably higher.
This may seem a silly example, but I think it actually highlights the important principle involved. In general when two bodies undergo an inelastic collision part of their kinetic energy is transferred to lattice vibrations i.e. heat. The second law of thermodynamics tells us that while transferring kinetic energy to heat is a highly probable process because the entropy change is positive, transferring heat back to kinetic energy is exceedingly improbable because the entropy change would be negative. In fact it's so improbable that it is effectively impossible.
But energy can be stored in other forms besides heat. The bullet stores chemical energy and there is no problem transferring chemical energy to heat because the entropy change when the hammer hits the bullet and the explosive goes off is large and positive. You could probably come with other scenarios e.g. a compressed spring that is released in the collision. All you need to arrange the collision such that the entropy change for an increase in kinetic energy is positive.
I will make my comments into an answer:
If we are talking of the classical domain, energy is conserved but can change forms. Inelastic scattering has a strict definition and is usually used describing two body scattering processes.
In two body impact situations conservation of input energy in the output products would require that the target has some hidden potential energy that can be released by the impact with the projectile. A ball made out of compressed springs hit hard enough by another hard ball may eject the springs and kinetic energy will be released. A ball of dynamite hit hard enough will explode and chemical energy will be released into kinetic energy.
When we enter the elementary particle scatterings then special relativity allows mass to turn into energy and thus proton antiproton inelastic scatters can have more kinetic energy than the impinging antiproton, as is evident with annihilations at rest, where the projectile has very little kinetic energy.