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Peter Gacs showed (Gacs 1974) that for every n there exists a bit string x oflength n whose plain complexity C(x) has almost maximal conditional complexityrelative to x, i.e., C(C(x)|x) > log n - log^(2) n - O(1). (Here log^(2) i =log log i.) Following Elena Kalinina (Kalinina 2011), we provide a simplegame-based proof of this result; modifying her argument, we get a better (andtight) bound log n - O(1). We also show the same bound for prefix-freecomplexity.  Robert Solovay showed (Solovay 1975) that infinitely many strings x havemaximal plain complexity but not maximal prefix complexity (among the stringsof the same length): for some c there exist infinitely many x such that |x| -C(x) < c and |x| + K(|x|) - K(x) > log^(2) |x| - c log^(3) |x|. In fact, theresults of Solovay and Gacs are closely related. Using the result above, weprovide a short proof for Solovay's result. We also generalize it by showingthat for some c and for all n there are strings x of length n with n - C (x) <c and n + K(n) - K(x) > K(K(n)|n) - 3 K(K(K(n)|n)|n) - c. We also prove a closeupper bound K(K(n)|n) + O(1).  Finally, we provide a direct game proof for Joseph Miller's generalization(Miller 2006) of the same Solovay's theorem: if a co-enumerable set (a set withc.e. complement) contains for every length a string of this length, then itcontains infinitely many strings x such that |x| + K(|x|) - K(x) > log^(2) |x|+ O(log^(3) |x|).

Complexity of complexity and strings with maximal plain and prefix Kolmogorov complexity