Pseudo-currying in one line

EDIT: Modified to cover situation when an argument is a List

Use Fold

expr = c[a1, a2, a3, a4, a5];

Fold[#1[#2] &, {c, List @@ expr} // Flatten[#, 1]&]

(*  c[a1][a2][a3][a4][a5]  *)

expr2 = c[a1, a2, {a31, a32, a33}, a4, a5];

Fold[#1[#2] &, {c, List @@ expr2} // Flatten[#, 1] &]

(*  c[a1][a2][{a31, a32, a33}][a4][a5]  *)

The deprecated (but valid) function HeadCompose basically does just that:

c[a1, a2, a3, a4, a5] /. h_[a___] :> HeadCompose[h, a]

c[a1][a2][a3][a4][a5]

If you don't wish to use that then perhaps one of these:

f1 = FixedPoint[Replace[h_[x_, y__] :> h[x][y]], #] &;

f2 = # //. {x : _[_] :> x, h_[x_, y__] :> h[x][y]} &;

f3 @ h_[x___, y_] := f3[h[x]][y]
f3 @ h_[] := h

e.g.

c[a1, a2, a3, a4, a5] // f1

c[a1][a2][a3][a4][a5]

Also possible (any maybe more readable) using patterns and ReplaceRepeated

c[a1, a2, a3, a4, a5]  //. f_[most__, last_] :> f[most][last]

c[a1][a2][a3][a4][a5]

As indicated by @MB1965 in a comment ReplaceRepeated is greedily searching for any part of the expression that matches f_[most__, last_] so that

c[a1 + a2, a3 + a4, a5] //. f_[most__, last_] :> f[most][last]

yields

c[a1[a2]][a3[a4]][a5]

Restricting the pattern to c[most__, last_] instead of f_[most__, last_] remedies that

pseudocurry[expr_] := expr //. Head[expr][most__, last_] :> Head[expr][most][last]

c[a1 + a2, a3 + a4, a5] // pseudocurry

c[a1 + a2][a3 + a4][a5]