Working in $ZF$ + existence of an inaccessible ordinal.

Let $\kappa$ be the first inaccessible ordinal (i.e. the first regular ordinal that is a limit of regular ordinals).

Let *subpartition* of $X$ stand for a "subset of a partition of $X$".

Question 1: Can we have a $V_{\kappa}$ sized subpartition of $\mathcal P(V_{\kappa})$ whose elements are $V_{\kappa}$ sized subpartitions of $V_{\kappa}$ whose elements are $V_{\kappa}$ sized?

Question 2:If so, can we go up? i.e. have a $V_{\kappa}$ sized subpartition of $\mathcal P^2(V_{\kappa})$ whose elements are $V_{\kappa}$ sized subpartitions of $\mathcal P(V_{\kappa})$ whose elements are $V_{\kappa}$ sized subpartitions of $V_{\kappa}$ whose elements are $V_{\kappa}$ sized?

Question 3:If so, then how long can we go up further? i.e. up to which finite ordinal $i$ we can have have a $V_{\kappa}$ sized subpartition of $\mathcal P^i(V_{\kappa})$ whose elements are $V_{\kappa}$ sized subpartitions of $\mathcal P^{i-1}(V_{\kappa})$,..., whose elements are $V_{\kappa}$ sized subpartitions of $V_{\kappa}$ whose elements are $V_{\kappa}$ sized?