Probably the most robust way is to store a linearly-growing
gameTime value that you can update by just adding the delta to it each frame, then set
gameSpeed = someConstant * pow(gameTime, 3.0) each frame.
But if you want to be able to do something interesting like changing the power continuously in real time, you could use the derivative: if y = t^n (in your case n = 3) then dy = nt^(n-1) dt = ny^((n-1)/n) dt, so you'd implement an update method like
power = 3.0; // Or whatever power you want; it could even change over time
gameSpeed += someConstant * delta * power * pow(gameSpeed, (power - 1.0) / power);
That should cause
gameSpeed to grow approximately as time cubed, but you could also alter the power to increase or decrease the rate of acceleration (or make it slow down again, using a negative power).