  I prove several theorems concerning upward closure and amalgamation in the
generic multiverse of a countable transitive model of set theory. Every such
model $W$ has forcing extensions $W[c]$ and $W[d]$ by adding a Cohen real,
which cannot be amalgamated in any further extension, but some nontrivial
forcing notions have all their extensions amalgamable. An increasing chain
$W[G_0]\subseteq W[G_1]\subseteq\cdots$ has an upper bound $W[H]$ if and only
if the forcing had uniformly bounded essential size in $W$. Every chain
$W\subseteq W[c_0]\subseteq W[c_1]\subseteq\cdots$ of extensions adding Cohen
reals is bounded above by $W[d]$ for some $W$-generic Cohen real $d$.
