ClearAll[y];
DSolve[{y''[t] == -(1/(y[t] - 1)), y[0] == 0, y'[0] == 0}, y[t], t]

{y[t] -> E^-InverseErf[-Sqrt[(2/[Pi])] t]^2 (-1 + E^
     InverseErf[-Sqrt[(2/[Pi])] t]^2)}

y[t] := E^-InverseErf[-Sqrt[(2/[Pi])] t]^2 (-1 + E^
     InverseErf[-Sqrt[(2/[Pi])] t]^2);
Plot[Evaluate[{y[t], 1/(y[t] - 1), D[y[t], t], D[y[t], {t, 2}]}], {t,
  0, 2}]

Solve[y[t] == 1, t]
{}

y[0.5]
y[0.5]

Evaluate[y[0.5]]
y[0.5]

y[t] /. t -> 0.5
0.12777