Actually your question perfectly illustrates the limitations of thinking of this as a computation device: I doubt it could be used to find the longest path, for the reason that whatever way the oil droplet responds at one intersection, it will employ the same type of response at the next.

The oil droplet is able to (usually) find the shortest path by always moving in the direction of higher acid content. Would you expect it to find the longest path by always going away from the higher acid content? That's wrong; that's not the longest path to the exit, it's the SHORTEST path back to the entrance!

OK, so how about the least acidic path that it hasn't already taken; i.e. not allow it to double back? Also wrong: that will lead it into the first blind alley that faces away from the exit, where it will be trapped.

So given that 1. We cannot use the highest acidic value as a criterion and 2. We cannot use the lowest acidic value as a criterion and 3. We cannot use the lowest acidic value not already visited, that leaves us with expecting the droplet to neither always take the most acidic exit from an intersection nor the least acidic. However, surely SOMETIMES the least acidic exit might be the longer path, and SOMETIMES the most acidic exit is. Then we are forced to conclude that the droplet must act differently at different times with the same stimulus!

That implies it has to maintain some kind of state, or memory. As the droplet is undifferentiated oil, it seems unlikely that it has a large enough state space to solve the longest path problem.