How to create a LR(k) grammar for an arbitrary k
Here's one pattern for an LR(k) grammar which is not LR(k-1).
I didn't fill in the definition of $A$; there's nothing particularly special about it. It might have an empty right-hand side, or it might match any LR(k) subgrammar. $a^k$ represents $k$ instances of $a$.

$$
\begin{align}S&amp;\to B a^{k-1} b\\
S&amp;\to C a^{k-1} c\\
B&amp;\to A\\
C&amp;\to A\\
A&amp;\to \text{see above}
\end{align}
$$

Clearly, it is not possible to determine whether $A$ should be reduced to $B$ or to $C$ without knowing what the $k$ following token is. So the grammar is not LR(i) for any $i \lt k$. The reduction can be determined using $k$ lookahead tokens, so the grammar is LR(k) provided that $A$ is.