"Inverse rc" impulse response circuit
This is a dangerous kind of want, as there has to be some sort of cutoff, you can't increase without bound (at least for long).

As far as replicating that behaviour, a real pole on the right side of the s-plane will have an step response of that nature. As for getting a pole there, it shouldn't be too hard, I don't think I've ever had a circuit naturally stable before compensating :P





MATLAB Code:

L = tf(1, [1 -10]) % real pole on right half plane
pzmap(L)
%%
step(L)
xlim([0 .5])   




As for another method (I'd recommend this one I think), the voltage to current relationship of a diode is exponential, meaning if you increase the forward voltage applied, the current increases proportional to a*e^(V/b), where a and b are relatively constant (both a and b will vary with temperature, something a high current can cause to rise, so be careful with power dissipation!). Look up diode current equation for the full picture.



Convert this current to voltage (transimpedance amp) and you will have an exponentially growing voltage in response to a linear increase in input.





If you want to cut out some of that long initial tail, just start sweeping the diode voltage at around 200mV.