Almost Gorenstein Rees algebras of $p_g$-ideals, good ideals, and powers of the maximal ideals

Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring and let $I$ be an ideal of $A$. We prove that the Rees algebra $\mathcal{R}(I)$ is an almost Gorenstein ring in the following cases: (1) $(A,\mathfrak{m})$ is a two-dimensional excellent Gorenstein normal domain over an algebraically closed field $K\cong A\mathrm{/}\mathfrak{m}$ and $I$ is a $p_g$-ideal; (2) $(A,{\mathfrak m})$ is a two-dimensional almost Gorenstein local ring having minimal multiplicity and $I={\mathfrak{m}}^{\mathrm{\ell }}$ for all $\mathrm{\ell }\ge 1$; (3) $(A,\mathfrak{m})$ is a regular local ring of dimension $d\ge 2$ and $I={\mathfrak{m}}^{d-1}$. Conversely, if $\mathcal{R}({\mathfrak{m}}^{\mathrm{\ell }})$ is an almost Gorenstein graded ring for some $\mathrm{\ell }\ge 2$ and $d\ge 3$, then $\mathrm{\ell }=d-1$.