We study stationary max-stable processes {n(t): t is an element of R} admitting a representation of the form n(t) = max(i is an element of N) (U-i +Y-i(t)), where Sigma(infinity)(i=1) delta U-i is a Poisson point process on R with intensity e(-u)du, and Y1,Y2 are i.i.d. copies of a process {Y (t) : t is an element of R} obtained by running a Levy process for positive t and a dual Levy process for negative t. We give a general construction of such Levy-Brown-Resnick processes, where the restrictions of Y to the positive and negative half-axes are Levy processes with random birth and killing times. We show that these max-stable processes appear as limits of suitably normalized pointwise maxima of the form Mn(t) = max(i=1),...,n xi i (s(n) + t), where xi 1, xi 2... are i.i.d. Levy processes and sn is a sequence such that s(n) similar to clog n with c > 0. Also, we consider maxima of the form max(i=1),...,n Z(i) (t/log n), where Zi, Z2,... are i.i.d. Ornstein-Uhlenbeck processes driven by an alpha-stable noise with skewness parameter beta = -1. After a linear normalization, we again obtain limiting max-stable processes of the above form. This gives a generalization of the results of Brown and Resnick (1977) to the totally skewed a-stable case. (C) 2015 Elsevier B.V. All rights reserved.