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 For[{n = 1, y[0][x_] = 1}, n < 4, n++, 
 y[n][x_] = 1 + Integrate[y[n - 1][t]^2 + t^2, {t, 0, x}];
 Print[{n, y[n][t]}]]

here how can I get table of values for t=0.1, 0.2, 0.3,.....,1?

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Is this what you are looking for?

y[0,x_] := 1;
y[n_,x_] := 1 + Integrate[y[n - 1,x]^2 + t^2, {t, 0, x}]
Table[{n,x,y[n,x]},{n,1,3},{x,.1,1,.1}]

which gives you

{{{1, 0.1, 1.10033}, {1, 0.2, 1.20267}, {1, 0.3, 1.309}, {1, 0.4, 1.42133}, {1, 0.5, 1.5416},
  {1, 0.6, 1.672},   {1, 0.7, 1.81433}, {1, 0.8, 1.97067}, {1, 0.9, 2.143}, {1, 1., 2.33333}},
 {{2, 0.1, 1.12141}, {2, 0.2, 1.29195}, {2, 0.3, 1.52304}, {2, 0.4, 1.82941}, {2, 0.5, 2.23003},
  {2, 0.6, 2.74935}, {2, 0.7, 3.41860}, {2, 0.8, 4.27749}, {2, 0.9, 5.37620}, {2, 1., 6.77778}}, 
 {{3, 0.1, 1.12609}, {3, 0.2, 1.33649}, {3, 0.3, 1.70490}, {3, 0.4, 2.36003}, {3, 0.5, 3.52819},
  {3, 0.6, 5.60736}, {3, 0.7, 9.29510}, {3, 0.8, 15.80819}, {3, 0.9, 27.25621}, {3, 1., 47.27160}}}

Check that carefully to make certain it is correct

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