The Internet routing ecosystem is facing compelling scalability challenges, manifested primarily in the rapid growth of IP packet forwarding tables. The forwarding table, implemented at the data plane fast path of Internet routers to drive the packet forwarding process, currently contains about half a million entries and counting. Meanwhile, it needs to support millions of complex queries and updates per second. In this paper, we make the curious observation that the entropy of IP forwarding tables is very small and, what is more, seems to increase at a lower pace than the size of the network. This suggests that a sophisticated compression scheme may effectively and persistently reduce the memory footprint of IP forwarding tables, shielding operators from scalability matters at least temporarily. Our main contribution is such a compression scheme which, for the first time, admits both the required information-theoretical size bounds and attains fast lookups, thanks to aggressive level compression. Although we find the underlying optimization problem NP-complete, we can still give a lightweight heuristic algorithm with firm approximation guarantees. This allows us to squeeze real IP forwarding tables, comprising almost 500, 000 prefixes, to just about 140 - 200 KBytes of memory within a factor of 2 - 3 of the entropy bound, so that forwarding decisions take only 8 - 10 memory accesses on average and updates are supported efficiently. Our compression scheme may be of more general interest, as it is applicable to essentially any prefix tree.