Consider a syntactic restriction of the (untyped) $\lambda$-calculus in which an application cannot have another application as an immediate subterm. More precisely, restricted terms ($R,S,...$) and non-application restricted terms ($N,O,...$) are given mutually recursively by the grammar: $$ R, S, ... ::= N \,\mid\, N\,N \\ N, O, ... ::= x \,\mid\, \lambda x.N $$ Equivalently: $$ R, S, ... ::= x \,\mid\, \lambda x. R \,\mid\, x\,y \,\mid\, x\,(\lambda y.R) \,\mid\, (\lambda x.R)\,y \,\mid\, (\lambda x.R)\,(\lambda y.S) $$ It can be checked that restricted terms are closed by $\beta$-reduction under arbitrary contexts.
What is the expressive power of this restricted $\lambda$-calculus (with $\beta$-reduction)? Can it express all computable functions? Remark that this restriction is not strong enough to guarantee termination; e.g. the non-terminating term $\Omega := (\lambda x. x\,x)\,(\lambda x. x\,x)$ is a restricted term.