Let $T: X\to X$ be a continuous map, where $X$ is a topological space. Fix also a family $\mathsf{I}\subseteq 𝒫(\mathbb{N})$ of "small" sets of nonnegative integers (for instance, the family $\mathrm{Fin}$ of finite sets, or the family $\mathsf{Z}$ of asymptotic density zero sets). A point $x \in X$ is said to be $\mathsf{I}$-hypercyclic if 
$$
\{n \in \mathbb{N}: T^nx \in U\}\notin \mathsf{I}
$$
for each nonempty open $U\subseteq X$. On a similar direction, a point $x \in X$ is said to be $\mathsf{I}$-strong hypercyclic if for each $y \in X$ there exists a subsequence $(T^nx: n \in A)$ of its orbit which is convergent to $y$ and, in addition, the set of indexes $A$ is \textquotedblleft not small,\textquotedblright\, that is, $A\notin \mathsf{I}$. In both cases, if $\mathsf{I}=\mathrm{Fin}$ and $X$ is a sufficiently nice topological space, then $x$ is $\mathsf{I}$-hypercyclic iff it is $\mathsf{I}$-strong hypercyclic iff its orbit is dense. 

We provide several structural relationships between the above notions and the related ones of recurrence with respect to $\mathsf{I}$. None of our results relies on the full linearity of $T$. As applications, we show that if $T$ is a homomorphism on a Fréchet space $X$ and there exists a dense set of vectors with orbits convergent to $0$, then for each $y \in X$ the set of all vectors $x \in X$ such that $\lim_{n \in A}T^nx=y$ for some $A\notin \mathsf{Z}$ is either empty or comeager. In a special case of bounded linear operators on Banach spaces, we obtain that $T$ is $\mathsf{Z}$-hypercyclic if and only if there exists a hypercyclic vector $x \in X$ for which $\lim_{n \in A}T^nx=0$ for some $A\notin \mathsf{Z}$. We conclude with several open questions.