More explicitly, $X$ is precisely $k$-connected if every continuous map to $X$ from the $k$-sphere extends to a continuous map to $X$ from the $k$-disk. Then $X$ is $n$-(simply) connected if $X$ is precisely $k$-connected for $-1 \leq k \leq n$.

Special cases

Any space is $(-2)$-simply connected.

A space is $(-1)$-simply connected precisely if it has an element; that is if it is inhabited.

A space is $0$-simply connected precisely if it is path-connected.

A space is $1$-simply connected precisely if it is simply connected.

A space is $\infty$-simply connected precisely if it is weakly contractible.

Terminology

The traditional terminology is ‘$n$-connected’, but this violates the rule that ‘$1$-foo’ should mean the same as ‘foo’. This can be fixed by saying ‘$n$-simply connected’ instead, which also has the advantage of stressing that we are extending the change from connected to simply connected spaces.