Q[x_] = (x^2)+c

ContourPlot[{Q[Q[Q[x]]]==x},{c,-2.1,-1.5},{x,-2,2},ContourStyle-> {{Red, Thickness[0.005]}},PlotRange-> All, Axes->True, Frame->False, AspectRatio->1]

NSolve[Q[Q[Q[0]]]==0,c,WorkingPrecision->15]

Qc[x_] = (x^2)-1.75487766624669272452763379988;

Qc[0]

Qc[Qc[0]]

Qc[Qc[Qc[0]]]

Plot[{x,Qc[x],Qc[Qc[Qc[x]]]},{x,-3,3},PlotRange->{{-3,3},{-3,3}},PlotStyle->{{Black, Thickness[0.01]},{Blue, Thickness[0.01]},{Red,Thickness[0.01]}}]

NSolve[x==Qc[Qc[Qc[x]]],x,WorkingPrecision->15]

Qc[-1.74285453783339449224376973661`15.]

Qc[Qc[-1.74285453783339449224376973661`15.]]

Qc[Qc[Qc[-1.74285453783339449224376973661`15.]]]

Qc[-0.91593702764165777166158855734`15.]

Plot[{x,Qc[Qc[Qc[x]]]},{x,1.25,1.4},PlotRange->{{1.25,1.4},{1.25,1.4}},PlotStyle->{{Black, Thickness[0.01]},{Red,Thickness[0.01]}}, AspectRatio->1]

Plot[{x,Qc[Qc[Qc[x]]]},{x,-0.15,0.15},PlotRange->{{-0.15,0.15},{-0.15,0.15}},PlotStyle->{{Black, Thickness[0.01]},{Red,Thickness[0.01]}}, AspectRatio->1]

Plot[{x,Qc[Qc[Qc[x]]]},{x,-1.8,-1.7},PlotRange->{{-1.8,-1.7},{-1.8,-1.7}},PlotStyle->{{Black, Thickness[0.01]},{Red,Thickness[0.01]}}, AspectRatio->1]

Plot[{x,Qc[x],Qc[Qc[Qc[x]]]},{x,-3,3},PlotRange->{{-3,3},{-3,3}},PlotStyle->{{Black, Thickness[0.001]},{Blue, Thickness[0.001]},{Red,Thickness[0.001]}}]