  We consider elliptic surfaces $\mathcal{E}$ over a field $k$ equipped with
zero section $O$ and another section $P$ of infinite order. If $k$ has
characteristic zero, we show there are only finitely many points where $O$ is
tangent to a multiple of $P$. Equivalently, there is a finite list of integers
such that if $n$ is not divisible by any of them, then $nP$ is not tangent to
$O$. Such tangencies can be interpreted as unlikely intersections. If $k$ has
characteristic zero or $p>3$ and $\mathcal{E}$ is very general, then we show
there are no tangencies between $O$ and $nP$. We apply these results to
square-freeness of elliptic divisibility sequences and to geography of
surfaces. In particular, we construct mildly singular surfaces of arbitrary
fixed geometric genus with $K$ ample and $K^2$ unbounded.
