Two approaches with Replacement.
z = Nest[1 + x^n (#) &, 1, 3] ReplacePart[z, Position[z, n][[#]] -> Defer@# & /@ Range[3]] 
Also inspired by the @Karsten7's answeranswer and @Eldo's Module version:
Module[{r = 3, i = 1}, Quiet@ReplaceRepeated[(1 + x^n), (1 + x^n) -> 1 + x^n (1 + x^n), MaxIterations -> r - 1] /. n :> i++]
1 + x (1 + x^2 (1 + x^3))
Two approaches with Replacement.
z = Nest[1 + x^n (#) &, 1, 3] ReplacePart[z, Position[z, n][[#]] -> Defer@# & /@ Range[3]]
Also inspired by the @Karsten7's answer and @Eldo's Module version:
Module[{r = 3, i = 1}, Quiet@ReplaceRepeated[(1 + x^n), (1 + x^n) -> 1 + x^n (1 + x^n), MaxIterations -> r - 1] /. n :> i++]
1 + x (1 + x^2 (1 + x^3))
Two approaches with Replacement.
z = Nest[1 + x^n (#) &, 1, 3] ReplacePart[z, Position[z, n][[#]] -> Defer@# & /@ Range[3]]
Also inspired by the @Karsten7's answer and @Eldo's Module version:
Module[{r = 3, i = 1}, Quiet@ReplaceRepeated[(1 + x^n), (1 + x^n) -> 1 + x^n (1 + x^n), MaxIterations -> r - 1] /. n :> i++]
1 + x (1 + x^2 (1 + x^3))
Another approach is to replace n's with respective numbers from Range[3]Two approaches with Replacement.
z = Nest[1 + x^n (#) &, 1, 3] 
Use ReplacePart with Map(/@)
ReplacePart[z, Position[z, n][[#]] -> Defer@# & /@ Range[3]]
If you do not want x^1 dropAlso inspired by the @Karsten7's Defer@.answer and @Eldo's Module version:
Module[{r = 3, i = 1}, Quiet@ReplaceRepeated[(1 + x^n), (1 + x^n) -> 1 + x^n (1 + x^n), MaxIterations -> r - 1] /. n :> i++]
1 + x (1 + x^2 (1 + x^3))
Another approach is to replace n's with respective numbers from Range[3].
z = Nest[1 + x^n (#) &, 1, 3]
Use ReplacePart with Map(/@)
ReplacePart[z, Position[z, n][[#]] -> Defer@# & /@ Range[3]]
If you do not want x^1 drop Defer@.
Two approaches with Replacement.
z = Nest[1 + x^n (#) &, 1, 3] ReplacePart[z, Position[z, n][[#]] -> Defer@# & /@ Range[3]]
Also inspired by the @Karsten7's answer and @Eldo's Module version:
Module[{r = 3, i = 1}, Quiet@ReplaceRepeated[(1 + x^n), (1 + x^n) -> 1 + x^n (1 + x^n), MaxIterations -> r - 1] /. n :> i++]
1 + x (1 + x^2 (1 + x^3))
Another approach is to replace n's with respective indexnumbers from Range[3].
sz = Nest[1 + x^n (#) &, 1, 3]
z = 1 + x^n (1 + x^n (1 + x^n)) Use ReplacePart with Map(/@)
ReplacePart[z, Position[z, n][[#]] -> Defer@# & /@ Range[3]]
If you do not want x^1 drop DeferDefer@.
Another approach is to replace n's with respective index.
s = Nest[1 + x^n (#) &, 1, 3]
z = 1 + x^n (1 + x^n (1 + x^n)) Use ReplacePart with Map(/@)
ReplacePart[z, Position[z, n][[#]] -> Defer@# & /@ Range[3]]
If you do not want x^1 drop Defer.
Another approach is to replace n's with respective numbers from Range[3].
z = Nest[1 + x^n (#) &, 1, 3]
Use ReplacePart with Map(/@)
ReplacePart[z, Position[z, n][[#]] -> Defer@# & /@ Range[3]]
If you do not want x^1 drop Defer@.