the algebraic k-theory of the category of finite type $n$ spectra is a fundamental object containing structural information about the stable homotopy category. however, until recently almost nothing was known about it for $n>1$, primarily because it is not the k-theory of a connective ring. in this talk, i will explain how, for $n=2$, it can be computed in terms of k-theory of discrete rings and topological cyclic homology. in particular, we can read off the k groups in low degrees and find that there is an infinite family of 2-torsion classes in $k_0$ at the prime 2. i will also explain how to construct type 2 spectra representing these $k_0$ classes.