f[n_] := n + Total[IntegerDigits[n]];

NestList[f, 1, 100]

NestList[f, 44, 100]

ListPlot[{NestList[f, 1, 100000], Table[5.25n Log[10, n], {n, 100000}]}] // Rasterize

N[NestList[f, 1, 1000-2] / Table[n Log[10, n], {n,2, 1000}]]

NestWhile[f, 1, (#<12345)&]

$RecursionLimit=123456;
fa[m_, n_] := fa[m, n] = If[n>m, n, fa[m, f[n]]]

fa[12345,1]

Union[fa[12345, #]& /@ Range[10000]]

fa[12345, #]& /@ Range[100]

Clear[f1, fa1];
f1[n_] := 22+n + Total[IntegerDigits[n]];
fa1[m_, n_] := fa1[m, n] = If[n>m, n, fa1[m, f1[n]]]

Union[fa1[1234567, #]& /@ Range[1000]]