In several iron-arsenide superconductors there is strong evidence for a fully gapped superconducting state consistent with either a conventional s-wave symmetry or an unusual s{+/-} state where the gap changes sign between the electron and hole Fermi-surface sheets. Here we report measurements of the penetration depth lambda(T) in very clean samples of the related iron-phosphide superconductor, LaFePO, at temperatures down to approximately 100 mK. We find that lambda(T) varies approximately linearly with T strongly suggesting the presence of gap nodes in this compound. Taken together with other data, this suggests the gap function is not universal for all pnictide superconductors.