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Refining a constructive combinatorial method due to MacLane and Schilling, wegive several criteria for a valued field that guarantee that all of its maximalimmediate extensions have infinite transcendence degree. If the value group ofthe field has countable cofinality, then these criteria give the sameinformation for the completions of the field. The criteria have applications tothe classification of valuations on rational function fields. We also apply thecriteria to the question which extensions of a maximal valued field, algebraicor of finite transcendence degree, are again maximal. In the case of valuedfields of infinite $p$-degree, we obtain the worst possible examples ofnonuniqueness of maximal immediate extensions: fields which admit an algebraicmaximal immediate extension as well as one of infinite transcendence degree.

Algebraic independence of elements in immediate extensions of valued fields