What is the free symmetric monoidal $\infty$-category on one object? It is well-known that the free symmetric monoidal category on one object is the category $\mathbb{F}$ of finite sets and bijections. This is supposed to be the categorification of the monoid of natural numbers, and its algebraic $K$-theory is given by the sphere spectrum $\mathbb{S}$, the free symmetric monoidal $\infty$-groupoid with inverses.
Is there a natural description of the free symmetric monoidal $\infty$-groupoid (resp. $\infty$-category, $(\infty,\infty)$-category) on one object?
 A: Yes, it is the same as $\mathbb{F}$.
As John Baez points out, it is the same as the free symmetric monoidal $\infty$-groupoid on one object. (This can also be seen by playing around with the adjoints between groupoids and categories).
Symmetric monoidal $\infty$-groupoids are the same as $E_\infty$-spaces (careful: not assumed to be "grouplike"). So we want the free $E_\infty$-space on a point. As a space this is homotopy equivalent to $\sqcup_{n\geq0} B \Sigma_n$. Note all the hom spaces are actually 1-types, so it can be modeled as a 1-groupoid, namely $\mathbb{F}$. It might seem surprising at first that the free symmetric monoidal $\infty$-category is actually an ordinary category. However on reflection we see that it follows from the fact that the spaces in the $E_\infty$-operad are actually 1-types.
Equivalently, as John says, it is the space of all finite subsets of $\mathbb{R}^\infty$. If you pick an identification of $\mathbb{R}^\infty$ with the "interior" of an infinite cube, then you get a natural $E_\infty$ structure on this model of the space, too.
You can also see that it is $\mathbb{F}$ in the way that Denis suggests in his comment to the OP.
