We consider tweakable blockciphers with beyond the birthday bound security. Landecker, Shrimpton, and Terashima (CRYPTO 2012) gave the first construction with security up to O(2(2n/3)) adversarial queries (n denotes the block size in bits of the underlying blockcipher), and for which changing the tweak does not require changing the keys for blockcipher calls. In this paper, we extend this construction, which consists of two rounds of a previous proposal by Liskov, Rivest, and Wagner (CRYPTO 2002), by considering larger numbers of rounds r > 2. We show that asymptotically, as r increases, the resulting tweakable blockcipher approaches security up to the information bound, namely O(2 n) queries. Our analysis makes use of a coupling argument, and carries some similarities with the analysis of the iterated Even-Mansour cipher by Lampe, Patarin, and Seurin (ASIACRYPT 2012).