It is well known that the weak (1, 1) bounds doesn't hold for the strong maximal operators, but it still enjoys certain weak L log L type norm inequality. Let Φ n (t) = t(1 + (log + t) n−1 ) and the space L Φn (R n ) be the set of all measurable functions on R n such thatIn this paper, we introduce a new weak norm space L 1,∞ Φn (R n ), which is more larger than L 1,∞ (R n ) space, and establish the correspondng limiting weak type behaviors of the strong maximal operators. As a corollary, we show that max 2 n ((n − 1)!) −1 , 1 is a lower bound for the best constant of the L Φn → L 1,∞ Φn norm of the strong maximal operators. Similar results have been extended to the multilinear strong maximal operators.