  We examine the Pythagoras number $\mathcal{P}(\mathcal{O}_K)$ of the ring of
integers $\mathcal{O}_K$ in a totally real biquadratic number field $K$. We
show that the known upper bound $7$ is attained in a large and natural infinite
family of such fields. In contrast, for almost all fields $\mathbb{Q}(\sqrt5,
\sqrt{s})$ we prove $\mathcal{P}(\mathcal{O}_K)=5$. Further we show that $5$ is
a lower bound for all but seven fields $K$ and $6$ is a lower bound in an
asymptotic sense.
