Many string theorists and cosmologists have recently turned their attention to building and testing string theory models of inflation. One of the main goals is to find novel features that could distinguish stringy models from their field theoretic counterparts. This is difficult because, in most examples, string theory is used to derived an effective theory operating at energies well below the string scale. However, since string theory provides a complete description of dynamics also at higher energies, it may be interesting to construct inflationary models which take advantage of this distinctive feature. I will discuss recent progress in this direction using p-adic string theory - a toy model of the bosonic string for which the full series of higher dimensional operators is known explicitly - as a playground for studying string cosmology to all order in $alpha\'$. The p-adic string is a nonlocal theory containing derivatives of all orders and this structure is also ubiquitous in string field theory. After discussing the difficulties (such as ghosts and classical instabilities) that arise in working with higher derivative theories I will show how to construct generic inflationary models with infinitely many derivatives. Novel features include the possibility of realizing slow roll inflation with a steep potential and large nongaussian signatures in the CMB.