ClearAll["Global`*"];
f[x_]:=x^2-2;
g[x_]:=x-f[x]/f'[x];
ng[x_]:=N[g[x],140];
n=10;
(v=NestList[ng,1,n])//TableForm;
ukaz[x_,y_]:=Row[
w1=ToString[x];
w2=ToString[y];
l1=StringLength[w1];
l2=StringLength[w2];
l =Min[l1,l2];
s1=StringTake[w1,l];
s2=StringTake[w2,l];
shoda=True;
t=Table[
c1=StringTake[s1,{i}];
c2=StringTake[s2,{i}];
color=If[c1==c2 && shoda,Red,shoda=False;Black];
cc=Style[c1,color],
{i,l}]];
Table[ukaz[v[[i]],v[[i+1]]],{i,n-1}]//TableForm