Partition-free subsets of $2^{[n]}$

Let $[n]$ denote the set of integers $\{1,2,\ldots,n\}$. A subset of $2^{[n]}$ is partition-free if it does not contain a partition of $[n]$.

What is the maximum size of a partition-free subset of $2^{[n]}$?

Note that it is easy to get such a subset of size $2^{n-1}$: for some choice $x\in[n]$, we have a partition-free subset $\{S : S\in 2^{[n]},\ x\notin S\}$.

Given a set $S$ and its complement, you can use at most one, so you can't do better than $2^{n-1}$.