This paper is the first in a series of three, about (relatively) free profinite semigroups and S-adic representations of minimal shift spaces. We associate to each primitive S-adic directive sequence σ a profinite image in the free profinite semigroup over the alphabet of the induced minimal shift space X(σ). When this profinite image contains a maximal subgroup of the free profinite semigroup, we say that σ is saturating. We show that if σ is recognizable, then it is saturating. Conversely, we use the notion of saturating sequence to obtain several sufficient conditions for σ to be recognizable: σ consists of pure encodings; or σ is eventually recognizable, saturating and consists of encodings; or σ is eventually recognizable, recurrent, bounded and consists of encodings. For the most part, we do not assume that σ has finite alphabet rank.