For example $(87,88)$ and $(607,608)$.
Kummer's theorem implies that, for all odd primes $p$, $p\mid \binom{2n}{n}$ if and only some base $p$ digit of $n$ is $>p/2$, and hence $(n,n+1)$ has the required property if for all primes $p\leq n$ we have $n\not\in \{\frac{p-1}{2},p-1\}\pmod{p}$. Standard heuristics then predict there should be \[\gg \frac{x}{(\log x)^2}\] many such $n\leq x$.
