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 $Q_n$?