Radicals will be studied for which the condition “A[x] ∈ λ for all nil rings A” is equivalent to the positive solution of K¨othe’s Problem (A[x] is Jacobson radical for all nil rings A, in Krempa’s formulation). The closer is to the Jacobson radical, the better approximation of the positive solution is obtained. Seeking, however, for a negative solution, possibly large radicals λ are of interest. In this note such large radicals will be studied.