What can be said about the set of primes $p$ for which it is proven that an infinite group with all nontrivial proper subgroups cyclic of order $p$ doesn't exist? Specifically, what is the largest such $p$ (say $p_0$)? All I could find in the literature is $p_0\le 10^{75}$, but I admit I didn't look very far...

$\begingroup$ Aren't there small primes p > 700 for which the finitely generated infinite Burnside groups exist? Or are you looking for p even and nonprime? Gerhard "Ask Me About System Design" Paseman, 2012.10.17 $\endgroup$ – Gerhard Paseman Oct 18 '12 at 5:35

$\begingroup$ Huh? p must be prime, as we are looking for "every subgroup" $\endgroup$ – Feldmann Denis Oct 18 '12 at 7:29

6$\begingroup$ I suspect that the largest prime for which this it is known that no Tarski Monster exists is 3 (but I might be wrong)! $\endgroup$ – Derek Holt Oct 18 '12 at 9:12
The largest known prime for which existence of Tarski monster is not known is $997$, see Adyan, S. I.; Lysënok, I. G. Groups, all of whose proper subgroups are finite cyclic. Izv. Akad. Nauk SSSR Ser. Mat. 55 (1991), no. 5, 933990; translation in Math. USSRIzv. 39 (1992), no. 2, 905–957. There are currently no methods of proving that for a given prime $p$ a Tarski monster does not exist except for showing that all finitely generated groups of exponent $p$ are finite. This gives hope that the bound $997$ can be lowered to $665$ (or below $300$ assuming the recent Adian's announcement is correct).