Let (X_i", Y_i) be a sequence of i.i.cl. random vectors where {X_i} are gains and {Y_i} are indicators of successes in repetitions of a game of heads and tails. Put, S₀ = 0, S_k = X₁+ ⋯ + X_k, and let I{.} denote the indicator function of the event in brackets. Then M_N = Max_0≦l<m≦N (S_m - S_l+1) I{Y_l+1N = ⋯ = Y_m = 1} is the maximal gain over sequences of successes without interruptions ("head runs"). We derive necessary and sufficient conditions for strong laws of large numbers for M_N and find rates of convergence in these laws.