Mathematica has a short tutorial entitled "Arbitrary-Precision Numbers." You can find it by searching for "tutorial/ArbitraryPrecisionNumbers" in Help/Wolfram Documentation or by typing SetPrecision, highlighting it, pressing F1, then find the link for the tutorial at the bottom of the page.

Hello again, sorry for the trouble. I've tried NDSolve method to search for "z" instead of ParametricNDSolveValue So here's what I done but still there's some error.

prog := Module[{Twater = 300., Tnew = 0, Cwater, Dwater, Ewater, 
    ClassicalRungeKuttaCoefficients, f, x, y, z = 60, point, ans, 
    tol = 0.001, iters = 0, iterLim = 100, prevTnew = 10^5, 
    plotList = {}}, $MinPrecison = 50;
   Cwater = 
    QuantityMagnitude[
     ThermodynamicData["Water", 
      "ThermalConductivity", {"Temperature" -> 
        Quantity[Twater, "Kelvins"]}]];
   ans = "Twinkle, twinkle, little star,
             How I wonder what you are.
             Up above the world so high,
             Like a diamond in the sky.
             Twinkle, twinkle, little star,
             How I wonder what you are!";

   While[(Abs[prevTnew - Tnew] > tol) && (iters++ < iterLim), 
    Dwater = (5 Cwater)/Pi;
    Ewater = Dwater^3 + 2*Cwater;

    While[
      (Abs[point] < 100) && (iters++ < iterLim),

      ClassicalRungeKuttaCoefficients[4, prec_] :=
       With[{amat = {{1/2}, {0, 1/2}, {0, 0, 1}}, 
         bvec = {1/6, 1/3, 1/3, 1/6}, cvec = {1/2, 1/2, 1}}, 
        N[{amat, bvec, cvec}, prec]];

      f = 
       NDSolve[{SetPrecision[
          Derivative[1][y][x] == 
           Piecewise[{{(y[x] + x^3 + 3 z - 120*Ewater), 
              0 <= x <= 1}, {(y[x] + x^2 + 2 z), 
              1 <= x <= 2}, {(y[x] + x + z), 2 <= x <= 3}}], 40], 
         y[0] == 0}, y, {x, 0., 3.}, 
        Method -> {"ExplicitRungeKutta", "DifferenceOrder" -> 4, 
          "Coefficients" -> ClassicalRungeKuttaCoefficients}, 
        StartingStepSize -> 1/100];

      point = y[1] /. f[[1]];

      z++

      ]

     Print[point];



    prevTnew = Tnew;

    Tnew = 170 + 1.89*z;

    Cwater = 
     QuantityMagnitude[
      ThermodynamicData["Water", 
       "ThermalConductivity", {"Temperature" -> 
         Quantity[Tnew, "Kelvins"]}]];

    (*Note Tnew here!*)
    Print["Twater: ", Twater, " Tnew: ", Tnew];];

   (*End while*)

   ans = {"Tnew: ", Tnew, "Cwater: ", Cwater, "Dwater: ", Dwater, 
     "Ewater: ", Ewater};
   Print["The answer is:\n", ans];];

What I tried to do is to have "While Loop" inside a "While Loop" and instead of FindRoot, I use ReplaceAll, Here's the plan,

The inner loop is to find the value of "z" so that the "point" is 100, and the increment of "z" is 1 each time. After it reach the correct value of "z", the the command will continue to look for Tnew, and repeat again.

But the problem is

 point = y[1] /. f[[1]];

does not work properly in the While loop. I can do it outside of While loop but not inside. So, i'm stuck here

All I was trying to show you was the basic paradigm for writing a program that finds an answer by iteration. There are several ways of doing this. As a former PhD student in Computer Science, I cannot imagine not having taken courses in numerical analysis (where iteration paradigms are in the first chapter), differential equations, and infinite series, plus many courses in business administration.

General Douglas MacArthur was ultimately Supreme Commander of Allied Forces in the Pacific Theater in World War II and military governor of Japan. He is widely credited, even in Japan, with being the father of modern Japan. His instructions from the American War Department were simple: "Make Japan look like the United States." In his autobiography, Reminisces, MacArthur makes the point over and over that he was a product of the education the Army afforded him. He showed again and again how this course or that course led directly to solutions to important problems.

I watched the videos on YouTube about the making of Fury, a movie about tank warfare in Europe in World War II. Fury's director said that the US sent thousands of young men to their deaths by burning them alive in those large metal boxes. I could not believe it. How is it possible that a culture that espouses progress, freedom for oneself and others, and concern for the poor kill so many of its young men in what was obviously a second rate weapon? A book on tank warfare shows it was simple. An isolationist population produced an isolationist congress that refused to fund much tank research; the US did not know how to make a good tank until the end of WW II. There are several other factors, not the least of which were lack of time and money, but one of the most important was that tank warfare doctrine was heavily influenced by men with a cavalry background, which said that the basic unit had to be light and fast. So, initially American tanks were light, fast, and prey to a single shot from a German anti-tank weapon or heavy panzer tank. My point is that the top American Ordnance and Armor planners simply could not escape their backgrounds to enable them to interpret the evidence supplied by the German blitzkrieg and the fall of France.

I have a point in writing this, and it is that you are obviously crippled by not knowing the rudiments of numerical analysis. But there are many other things you may not know: For instance, forecasting is nothing like naïve thinking might make one think. For another example, one of the most significant causes of American homelessness is a young person having a bad first boss who does not lead the young person to enjoy work. Do you know how to forecast or lead young people to be successful in life and work? If you are offered a chance to enter a PhD program, I urge you to seriously consider it. That or something similar is the only way to escape the tyranny of your impoverished background (numerical-analysis-wise). Also, everybody whoever made it had a mentor. Find someone in business, industry, or academia where you want to work who can lead you to study the issues they deal with now and will deal with in the future. The higher your mentor's rank, the higher yours will likely be.

I made the algorithm converge to an answer; I have no idea if it is correct. Here are the changes I made:

I replaced the statement Twater = Tnew, with Cwater = QuantityMagnitude[ ThermodynamicData["Water", "ThermalConductivity", {"Temperature" -> Quantity[Tnew, "Kelvins"]}]]; <-- note the Tnew here. I added the variable prevTnew, initialized it to 10^5, and set it to Tnew just before Tnew is computed. I changed the completion test to (Abs[prevTnew - Tnew] > tol) && (iters++ < iterLim). Also, please note I created the variable plotList, and initialized it to plotList = {}. Then after you compute point, I added AppendTo[plotList, point]. Later, you could put in Print[ListPlot[plotList]] if the reason you compute point is to plot it. I don't understand the 100 in point. Here is the new routine:

prog := Module[{Twater = 300., Tnew = 0, Cwater, Dwater, Ewater, 
    ClassicalRungeKuttaCoefficients, f, x, y, point, ans, tol = 0.001,
     iters = 0, iterLim = 100, prevTnew = 10^5, plotList = {}},
   $MinPrecison = 50; 
   Cwater = 
    QuantityMagnitude[
     ThermodynamicData["Water", 
      "ThermalConductivity", {"Temperature" -> 
        Quantity[Twater, "Kelvins"]}]];
   ans = "Twinkle, twinkle, little star,
         How I wonder what you are.
         Up above the world so high,
         Like a diamond in the sky.
         Twinkle, twinkle, little star,
         How I wonder what you are!";
   While[(Abs[prevTnew - Tnew] > tol) && (iters++ < iterLim), 
    Dwater = (5 Cwater)/Pi;
    Ewater = Dwater^3 + 2*Cwater;
    ClassicalRungeKuttaCoefficients[4, prec_] := 
     With[{amat = {{1/2}, {0, 1/2}, {0, 0, 1}}, 
       bvec = {1/6, 1/3, 1/3, 1/6}, cvec = {1/2, 1/2, 1}}, 
      N[{amat, bvec, cvec}, prec]];
    f = ParametricNDSolveValue[{SetPrecision[
        Derivative[1][y][x] == 
         Piecewise[{{(y[x] + x^3 + 3 z - 120*Ewater), 
            0 <= x <= 1}, {(y[x] + x^2 + 2 z), 
            1 <= x <= 2}, {(y[x] + x + z), 2 <= x <= 3}}], 40], 
       y[0] == 0}, y[3.], {x, 0., 3.}, z, 
      Method -> {"ExplicitRungeKutta", "DifferenceOrder" -> 4, 
        "Coefficients" -> ClassicalRungeKuttaCoefficients}, 
      StartingStepSize -> 1/10];
    point = {z /. FindRoot[f[z] == 100., {z, 1}, Evaluated -> False], 
      100.};
    AppendTo[plotList, point];
    Print[point];
    FindRoot[f[z] == 100., {z, 1}, Evaluated -> False];
    prevTnew = Tnew;
    Tnew = 
     170 + 1.89*z /. 
      FindRoot[f[z] == 100., {z, 1}, Evaluated -> False];
    Cwater = 
     QuantityMagnitude[
      ThermodynamicData["Water", 
       "ThermalConductivity", {"Temperature" -> 
         Quantity[Tnew, "Kelvins"]}]]; (* Note Tnew here! *)
    Print["Twater: ", Twater, " Tnew: ", Tnew];
    ];(*End while*)
   ans = {"Tnew: ", Tnew, "Cwater: ", Cwater, "Dwater: ", Dwater, 
     "Ewater: ", Ewater};
   Print["The answer is:\n", ans];
   ];

Hello again, I've tried the code you provided. The output of the code is

prog
{67.9545,100.}
Twater: 300 Tnew: 298.434
The answer is:
Twinkle, twinkle, little star.

The answer for Tnew is the answer for the first loop. The exact answer can be achieve at about 10 to 15 loops and Tnew = 295.565 I think I'm not fully understand how While[] loop works. From the documentation, While[test,body] is to run a number of tests on the body until it no longer holds true right? But the loop only run once....

I'm sorry I need to ask the questions here because this is the only place I can learn Mathematica

From the code give

While[(Abs[Twater - Tnew] > tol) && (iters++ < iterLim),

Does it means that the "test" of while is to check whether Twater-Tnew is more than tol (0.001), if it is yes, the continue the iteration by replace Twater with Tnew from the code at the end right?

 Twater = Tnew

Or there's something I missed?

The code below is the general method, but I am not sure it achieves the correct answer. Start it by "Shift+Enter" or "Keypad Enter" on the cell containing prog. You debug it by putting in print statements until you find the error.

prog := Module[{Twater = 300, Tnew = 0, Cwater, Dwater, Ewater, 
    ClassicalRungeKuttaCoefficients, f, x, y, point, ans, tol = 0.001,
     iters = 0, iterLim = 100},
   Cwater = 
    QuantityMagnitude[
     ThermodynamicData["Water", 
      "ThermalConductivity", {"Temperature" -> 
        Quantity[Twater, "Kelvins"]}]];
   ans = "Twinkle, twinkle, little star,
     How I wonder what you are.
     Up above the world so high,
     Like a diamond in the sky.
     Twinkle, twinkle, little star,
     How I wonder what you are!";
   While[(Abs[Twater - Tnew] > tol) && (iters++ < iterLim),
    Dwater = (5 Cwater)/Pi;
    Ewater = Dwater^3 + 2*Cwater;
    ClassicalRungeKuttaCoefficients[4, prec_] := 
     With[{amat = {{1/2}, {0, 1/2}, {0, 0, 1}}, 
       bvec = {1/6, 1/3, 1/3, 1/6}, cvec = {1/2, 1/2, 1}}, 
      N[{amat, bvec, cvec}, prec]];
    f = ParametricNDSolveValue[{Derivative[1][y][x] == 
        Piecewise[{{(y[x] + x^3 + 3 z - 120*Ewater), 
           0 <= x <= 1}, {(y[x] + x^2 + 2 z), 
           1 <= x <= 2}, {(y[x] + x + z), 2 <= x <= 3}}], y[0] == 0}, 
      y[3.], {x, 0., 3.}, z, 
      Method -> {"ExplicitRungeKutta", "DifferenceOrder" -> 4, 
        "Coefficients" -> ClassicalRungeKuttaCoefficients}, 
      StartingStepSize -> 1/10];
    point = {z /. FindRoot[f[z] == 100., {z, 1}, Evaluated -> False], 
      100.};
    Print[point];
    FindRoot[f[z] == 100., {z, 1}, Evaluated -> False];
    Tnew = 
     170 + 1.89*z /. 
      FindRoot[f[z] == 100., {z, 1}, Evaluated -> False];
    Print["Twater: ", Twater, " Tnew: ", Tnew];
    Twater = Tnew;
    ];  (* End while *)
   Print["The answer is:\n", ans];
   ];

prog