One of the fundamental trade-offs in compact routing schemes is between the space used to store the routing table on each node and the stretch factor of the routing scheme – the ratio between the cost of the route induced by the scheme and the cost of a minimum cost path between the same pair. Using a distributed Kolmogorov Complexity argument, we give a lower bound for the name-independent model that applies even to single-source schemes and does not require a girth conjecture. For any integer k ≥ 1 we prove that any routing scheme for networks with arbitrary weights and arbitrary node names (even a single-source routing scheme) with maximum stretch strictly less than 2k + 1 requires Ω((nlogn)1/k)-bit routing tables. We extend our results to lower bound the average-stretch, showing that for any integer k ≥ 1 any name-independent routing scheme with (n/(9k))1/k-bit routing tables has average-stretch of at least k/4 + 7/8. This result is in sharp contrast to recent results on the average-stretch of labeled routing schemes.