Sorry, I suddenly got busy and my reply was delayed.

I had never thought of a way to create temporary variables using Unique, but it seems like a good way. Although I'm not familiar with handling Context, I found that side effects can be cancelled by doing something like Remove["`MyPrivate`*"].

An advantage of my function notation is that zero or negative number "subscripts" are available, and this is the same with your proposed Subscript description.Additionally, it feels more natural and reasonable to use Subscript than function notation. It somehow doesn't be affected by N[]...

My question have been resolved, thank you!

I have made a slight modification to your code to make it work, but I still get the error output.

solveEquations3[a_, varCount_] := Module[
  {x, vars}
  ,
  vars = Table[x[i], {i, 0, varCount}];
  Values[
   FindRoot[
    Catenate[{{vars[[1]] == 1}, 
      Table[vars[[i]] == Gamma[a, vars[[i - 1]]], {i, 2, 
        varCount + 1}]}], Thread[{vars, 1}]]]]

solveEquations3[1., 3]

What problems do you see with Gianluca's solution?

Try SetAttributes[x, NHoldFirst].