To Infinity (but not beyond)

The inductor current ramp rate is proportional to the applied voltage. If the voltage is constant, so is the current ramp rate. Since $V=1V$ and $L = 1H$ , the ramp rate is $\frac{V}{L}=\frac{dI}{dt}=1A\cdot s^{-1}\rightarrow I = I_0 + (A\cdot s^{-1})t$

(Assuming $t$ is in seconds . . . MKS/SI units and all.)

Plug in the time $t=8675309s$ and we have the answer as $8675309A$ .

Note: In real life, you would fry the battery or power source before you can get this kind of current.

Here V=1 Volt and L=1 Henry. Therefore the equation for the growth of current is dI/dt=1 with the initial condition I(0)=0. Therefore I=t.