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The paper is devoted to classification of finite abelian extensions L/K which satisfy the condition [L:K]|D L/K. Here K is a complete discretely valued field of characteristic 0 with an arbitrary residue field of prime characteristic p, D L/K is the different of L/K. This condition means that the depth of ramification in L/K has its “almost maximal” value. The condition appeared to play an important role in the study of additive Galois modules associated with the extension L/K. The study is based on the use of the notion of independently ramified extensions, introduced by the authors. Two principal theorems are proven, describing almost maximally ramified extensions in the cases when the absolute ramification index e is (resp. is not) divisible by p-1. Bibliography: 7 titles.