We will discuss two weak versions of recursive saturation: K-tallness and tallness. K-tallness characterizes the countable models of IΔ₀ + exp that have cofinal elementary extensions enlarging any infinite definable set, and tallness characterizes such models of linear ordering without the last element. I will present the proof of the latter. If time permits, we will also discuss the two properties in relation to the existence of κ-pseudosaturated cofinal elementary extensions. Here, κ-pseudosaturated structures are the structures all of whose infinite definable sets have a cardinality ≥κ.