Obviously, all free Burnside groups $B(m,n)$ are recursively presentable, so examples with the uniform torsion are possible here. Of course, the proof is much more complicated than for Grigorchuk's group.

For finitely presented groups the question is opened, even for the case of unbounded exponents. The closest result is by I.Ivanov-Pogodaev and A.Kanel-Belov, that gives an example of finitely presented infinite nil-semigroup $\Pi_0$, i.e. a semigroup where every element is equal in some power to an element $0$, for which identities $0x=0$, $x0=x$ hold for every $x \in \Pi_0$.

Obviously, all free Burnside groups $B(m,n)$ are recursively presentable, so examples with the uniform torsion are possible here. Of course, the proof is much more complicated than for Grigorchuk's group.

For finitely presented groups the question is opened, even for the case of unbounded exponents. The closest result is by I.Ivanov-Pogodaev and A.Kanel-Belov, that gives an example of finitely presented infinite nil-semigroup $\Pi_0$, i.e. a semigroup where every element is equal in some power to an element $0$, for which identities $0x=0$, $x0=0$ hold for every $x \in \Pi_0$.