$(X_\alpha : \alpha < \omega_2) \subset \wp(\omega_1)$ is a strong chain in $\wp(\omega_1)$/Fin if and only if $X_\beta - X_\alpha$ is finite and $X_\alpha - X_\beta$ is uncountable for each $\beta < \alpha < \omega_1$. We show that it is consistent that a strong chain in $\wp(\omega_1)$ exists. On the other hand we show that it is consistent that there is a strongly almost-disjoint family in $\wp(\omega_1)$ but no strong chain exists: $\square_{\omega_1}$ is used to construct a c.c.c forcing that adds a strong chain and Chang's Conjecture to prove that there is no strong chain.