Topological Hochschild homology (THH) has been used successfully to construct prismatic cohomology, the most powerful cohomology theory that we currently have for formal schemes over Z_p. But for rigid-analytic varieties over Q_p, or even global objects like varieties over Q, THH is less useful: For rational inputs, THH will always vanish modulo p, and so it can't have any interesting comparisons to, say, étale cohomology with torsion coefficients. In this talk, I'll explain a refinement of THH (due to Efimov and Scholze) that overcomes these issues, and I'll show a few promising computations. I'll also sketch how one should be able to recover Scholze's analytic Habiro cohomology from refined THH.