We say that an element g of a group G is almost right Engel if there is a finite set R(g) such that for every x∈G all sufficiently long commutators [...[[g,x],x],…,x] belong to R(g), that is, for every x∈G there is a positive integer n(x,g) such that [...[[g,x],x],…,x]∈R(g) if x is repeated at least n(x,g) times. Thus, g is a right Engel element precisely when we can choose R(g)={1}. We prove that if all elements of a compact (Hausdorff) group G are almost right Engel, then G has a finite normal subgroup N such that G/N is locally nilpotent. If in addition there is a uniform bound |R(g)|≤m for the orders of the corresponding sets, then the subgroup N can be chosen of order bounded in terms of m. The proofs use the Wilson--Zelmanov theorem saying that Engel profinite groups are locally nilpotent and previous results of the authors about compact groups all elements of which are almost left Engel
