What can take kinetic energy, transform it into potential energy when pressed on, and put back out as kinetic energy when released (besides a spring)? A spring can only hold so much of the kinetic energy. For example, a 1 cm spring can hold less than 5 J. Is there anything that can hold a large amount of energy but be fairly small?
 A: A 1 kg object on the surface of the Earth can store up to $G\frac{M_e}{r_e}\cdot 1$ $kg = 6.3 \times 10^7$ $J$ in gravitational potential energy as long as your pressing direction is up and you press it up far enough.
A compressible thermodynamic system can theoretically hold an infinite amount of energy. See for example the apparati in this image, which are boxes with pistons: http://chemwiki.ucdavis.edu/@api/deki/files/10077/STEP_4-1.jpg
Assume for example that our box is initially 1 cm^3 (0.000001 m^3), and is initially filled with air (which we'll assume to be an ideal gas) at room temperature (~300 K), and atmospheric pressure (~100,000 Pa). Then this box will initially contain ~0.15 J of thermal energy. Then by the adiabatic relation for an ideal gas, which states that $PV^{5/3}$ is constant under any adiabatic process, and the (energy form of the) ideal gas law $\tfrac{3}{2}PV = U$ where $U$ is total internal energy, we can do some algebra and derive this formula for the change in energy as a function of added pressure (in SI units):
$\Delta U =\frac{3}{2}\cdot 0.001 \cdot (100000+\Delta P)^{2/5} - 0.15$ $J$.
Unfortunately, realistically this means you'd need to apply more than a million atmospheres of pressure in order to store more than a couple dozen Joules of energy, and there do not currently exist materials that can take those kind of pressures, as far as I know.
Finally, you could store the energy by compressing a charged parallel plate capacitor, but if the capacitor is fairly small then electrons are liable to jump from the negative end to the positive end when the plates get close enough, which will limit the amount of energy you can store.
