Definition: An element $x\in X_n$ is said to be spherical if $d_i x=*$ for all $0\leq i\leq n$. $X$ is a pointed fibrant simplicial set.
I am puzzling over this definition.
For instance, if $x=(x_0,x_1,\dots,x_n)$, then $d_i x=(x_0,\dots,x_{i-1},x_{i+1},\dots,x_n)$ is firstly different for each $i$, and secondly comprised of $n$ components so it can't possibly be the basepoint $*$?
Is there any concrete example of such a spherical element $x$?
It sounds like the question arises from two points of confusion.
1) If you think of * as an element of $X_0$, then you need to remember that an element of $X_0$ gives rise, via the degeneracy maps, to an element in each $X_n$. (Play around with the simplicial identities to see why you get a well-defined element in $X_n$, regardless of which degeneracy maps you compose to get from $X_0$ up to $X_n$.)
2) In a simplicial set, elements in $X_n$ are not uniquely determined by a list of vertices. And there is nothing preventing two faces of a simplex from being the same. In this question, the relevant point is that for some simplices, all faces are the same! Think of a triangle in which all three vertices have been glued together into one point. You can even go a step further and collapse all three edges onto that same point. Now you have a sphere!