  We propose topological Hochschild homology as a tool for measuring
ramification of maps of structured ring spectra. We determine second order
topological Hochschild homology of the $p$-local integers. For the tamely
ramified extension of the map from the connective Adams summand to $p$-local
complex topological K-theory we determine the relative topological Hochschild
homology and show that it detects the tame ramification of this extension. We
also determine relative topological Hochschild homology for the
complexification map from connective real to complex topological K-theory and
for some quotient maps with commutative quotients.
