Thanks everyone!
A version suggested by Toni Schindler:
Fold[#[#2] &, f, {a, b, c, d}]
f[a][b][c][d]
Dang! Jose beat me to it. :)
What about something like this?
In[1]:= Fold[Compose, f, {a, b, c, d}]
Out[1]= f[a][b][c][d]
Jose.
Nice one, @Jose M. Martin-Garcia! Docs say Compose has been superseded by Composition but simple replacement of former by later in your code did not work. @Jose M. Martin-Garcia, @Christopher Carlson, do you know what would do the trick?
Composition