Initially no current flows through the $600k\Omega$ resistor. It all flows onto the uncharged capacitor. After a long time the capacitor will be fully charged and the current through the $600k\Omega$ resistor is the same as that through the battery, which is $\frac{80}{3}\mu A$.
So the current through the resistor increases exponentially from $0$ to a limit of $$I_{\infty} = \frac{80}{3} \mu A$$ The rate of increase is set by the time constant of the parallel combination of resistor and capacitor, which is $\tau=CR$, where $R=600k\Omega$ and $C=2.5nF$. Such an increase can be written as $$I(t)=I_{\infty}(1-e^{-t/ \tau})$$