Each of the hobbits eats $\frac{1}{4}$ of the porridge in the pan, so leaves $\frac{3}{4}$ in place. This means that the amount of porridge is multiplied by $\frac{3}{4}$ each time one of the hobbits eats.
Since four of them eat, the amount of porridge is multiplied by $\frac{3}{4} \times \frac{3}{4} \times \frac{3}{4} \times \frac{3}{4} = \left( \frac{3}{4} \right)^4 = \frac{81}{256}$
Therefore, $\frac{81}{256}$ of the porridge remains.