Non-abelian Group with infinite exponent in which every proper subgroup has finite exponent can you find a Non-abelian Group with infinite exponent in which every proper subgroup has finite exponent? 
 A: They, do, indeed, exist - see Tarski monsters.
More precisely:
In the article

A. Yu. Olshanskii, An infinite group with subgroups of prime orders,
  Math. USSR Izv. 16 (1981), 279–289; translation of Izvestia Akad. Nauk
  SSSR Ser. Matem. 44 (1980), 309–321.

an infinite group $G$ is constructed such that every proper subgroup of $G$ has prime order (for different primes in general). Unfortunately, the article is by itself unreadable, as it relies heavily on some previous article of the author. 
By the same author, the following article

A. Yu. Olshanskii, Groups of bounded period with subgroups of prime
  order, Algebra and Logic 21 (1983), 369–418; translation of Algebra i
  Logika 21 (1982), 553–618.

concerns the existence of "honest" Tarski monsters, i.e. infinite group $G$ such that every proper subgroup is of order $p$ for a fixed prime $p$. However, as author states in the introduction:

The infinite non-Abelian groups whose proper subgroups are all finite, constructed in [...]
   Obviously, these groups are periodic
  but they contain elements of arbitrarily large orders. 

Thus, the groups constructed in the original article answer the question.
