Can a hereditary graph class contain almost all, but not all, n-vertex graphs? Let $Q$ be a hereditary class of graphs. (Hereditary = closed with respect to taking induced subgraphs.) Let $Q_n$ denote the set of $n$-vertex graphs in $Q$. Let us say that $Q$ contains almost all graphs, if the fraction of all $n$-vertex graphs falling in $Q_n$ approaches 1, as $n\rightarrow\infty$. 

Question: Is it possible that a hereditary graph class $Q$ contains almost all graphs, but for every $n$ there is at least one graph that is not in $Q_n$?
The answer is no - for a fixed $Q$ let $t$ be the number of vertices in the smallest graph $H$ not in $Q$. Now, consider $n$ much bigger than $t$. For a random graph on $n$ vertices, the probability that the $t$ first vertices induce $H$ depends only on $t$. Partitioning the vertex set into $n/t$ disjoint sets of size $t$ and considering the probability that none of the sets are equal to $H$ shows that the probability of being in $Q$ tends to $0$ as $n$ increases.