Many random media models involve random variables attached to sites in a state space such as $Z^d$ or $ R^d$. In these models, the sums of values of these random variables (or integrals) along paths through the state space are are of great interest. In many cases, the supremum or infimum of these sums are physically relevant. Examples of this are first passage percolation and the parabolic Anderson model. The growth of these extrema are generally linear in the path length and satisfy a law of large numbers. In this talk we examine deviations above the mean and below the mean for a variety of models and show they are generally very unbalanced.