= Solution
For a regular spherical star, let $\rho_c=\rho(0)$ and $P_c=P(0)$. The <stellar mass conservation equation> and <stellar hydrostatic equation> are
$$
M'(r)=4\pi r^2\rho(r),\qquad P'(r)=-\frac{GM(r)\rho(r)}{r^2}.
$$
Assuming the smoothness needed for the requested <Taylor series>, write $M=a_0+a_1r+a_2r^2+a_3r^3+\cdots$. There is no point <mass> at the centre, so $a_0=0$, and matching the <stellar mass conservation equation> gives $a_1=a_2=0$, $a_3=4\pi\rho_c/3$. Substitution in the <stellar hydrostatic equation> gives $P'(r)=-(4\pi G/3)\rho_c^2r+O(r^3)$, hence
$$
\boxed{M(r)=\frac{4\pi}{3}\rho_cr^3+O(r^5),\qquad
P(r)=P_c-\frac{2\pi G}{3}\rho_c^2r^2+O(r^4).}
$$
The remainder orders use a smooth spherically symmetric scalar <density>, for which $\rho(r)=\rho_c+O(r^2)$. Merely finite continuous <density> gives the same leading coefficients with $o(r^3)$ and $o(r^2)$ remainders. In particular $P'(0)=0$, despite the apparent $r^{-2}$ in the <stellar hydrostatic equation>.
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