It is shown that every transcendental entire function f grows transcendentally in a sequence of cercles de remplissage. An example shows that if

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then there may be no sequence of cercles de remplissage the union of which contains infinitely many zeros of f. It is also shown that every transcendental entire function f has a Hayman direction, that is, a direction θ such that, in every open sector containing θ, either f assumes all complex values infinitely often, or else every derivative of f assumes all complex values, except possibly zero, infinitely often.