  In this paper we study the following problem proposed by Barrus, Ferrara,
Vandenbussche, and Wenger. Given a graph $H$ and an integer $t$, what is
$\operatorname{sat}_{t}\left(n, \mathfrak{R}{(H)}\right)$, the minimum number
of edges in a $t$-edge-coloured graph $G$ on $n$ vertices such that $G$ does
not contain a rainbow copy of $H$, but adding to $G$ a new edge in any colour
from $\{1,2,\ldots,t\}$ creates a rainbow copy of $H$? Here, we completely
characterize the growth rates of $\operatorname{sat}_{t}\left(n,
\mathfrak{R}{(H)}\right)$ as a function of $n$, for any graph $H$ belonging to
a large class of connected graphs and for any $t\geq e(H)$. This classification
includes all connected graphs of minimum degree $2$. In particular, we prove
that $\operatorname{sat}_{t}\left(n, \mathfrak{R}{(K_r)}\right)=\Theta(n\log
n)$, for any $r\geq 3$ and $t\geq {r \choose 2}$, thus resolving a conjecture
of Barrus, Ferrara, Vandenbussche, and Wenger. We also pose several new
problems and conjectures.
