optsall = {Axes -> False, Frame -> True, ImageSize -> 600,
   BaseStyle -> {Thick, FontSize -> 14}};
opts1 = {ImagePadding -> {{Scaled[0.05], Scaled[0.11]}, {Scaled[0.04],
       Scaled[0.02]}},
   FrameStyle -> {Thick, {Thick, Blue}, Thick, Opacity[0]},
   FrameLabel -> {"x", "First y-label"}};
opts2 = {ImagePadding -> {{Scaled[0.05], Scaled[0.11]}, {Scaled[0.04],
       Scaled[0.02]}},
   FrameStyle -> {Opacity[0], Opacity@0, Opacity@0, Green},
   FrameTicks -> All,
   FrameLabel -> ConstantArray["Second y-label", 4]};
opts3 = {ImagePadding -> {{Scaled[0.11], Scaled[0.05]}, {Scaled[0.04],
       Scaled[0.02]}},
   FrameStyle -> {Opacity[0], Opacity@0, Opacity@0, Red},
   FrameTicks -> All, FrameLabel -> ConstantArray["Third y-label", 4]};
Overlay[{
  Plot[{x, Rescale[x^2, {0, 100}, {0, 10}],
      Rescale[x^3, {0, 1000}, {0, 10}]}, {x, 0, 10},
     PlotStyle -> {Blue, Green, Red},
     ##, PlotLegends ->
      Placed[LineLegend[{"y = x", "y = x^2", "y = x^3"},
        LegendFunction -> (Framed[#] &)], Scaled[{0.15, 0.8}]]] & @@
   Join[optsall, opts1],
  Plot[x^2, {x, 0, 10}, PlotStyle -> None,
     ##] & @@ Join[optsall, opts2],
  Plot[x^3, {x, 0, 10}, PlotStyle -> None, ##] & @@
   Join[optsall, opts3]

  }]

Clear[threeyaxisplot];
threeyaxisplot[{f1_, f1label_}, {f2_, f2label_}, {f3_, f3label_},
  range : {var_Symbol, vmin_?NumericQ, vmax_?NumericQ}] :=
 Module[{plot1, plot2, plot3, p1yrng, p2yrng, p3yrng, p2yaxis,
   p3yaxis},
  {
   {plot1, plot2, plot3} =
    Plot[#[var], range, Axes -> False,
       Frame -> {True, True, False, False},
       PlotRangeClipping -> False] & /@ {f1, f2, f3}
   };
  {p2yaxis, p3yaxis} =
   Plot[(First@#)[var], range, Axes -> False,
      Frame -> {False, True, False, False}, PlotStyle -> White,
      FrameLabel -> {None, Last@#}] & /@ {{f2, f2label}, {f3,
      f3label}};
  {p1yrng, p2yrng, p3yrng} =
   Last@(PlotRange /. Options[#, PlotRange]) & /@ {plot1, plot2,
     plot3};
  Plot[{f1[var], Rescale[f2[var], p2yrng, p1yrng],
    Rescale[f3[var], p3yrng, p1yrng]}, range,
   PlotRangeClipping -> False,
   Axes -> False, Frame -> {True, True, False, False},
   FrameLabel -> {None, f1label},
   Epilog -> {
     Inset[p2yaxis, Scaled@{1.6, 0.5}, Scaled@{0.5, 0.5}, Scaled@1.07],
     Inset[p3yaxis, Scaled@{1.85, 0.5}, Scaled@{0.5, 0.5},
      Scaled@1.07]},
   ImagePadding -> {{Automatic, 130}, {Automatic, Automatic}}]
  ]
out = threeyaxisplot[{# &, "function1"}, {#^2 &, "function2"}, {#^3 &,
    "function3"}, {x, 0, 10}]