Is this a bug or do I don't know enough about `N`:
f[x_] = Piecewise[{{2 x, 0 <= x < 1/2}, {2 x - 1, 1/2 <= x <= 1}}];
Block[{$MaxExtraPrecision = 500},
neworbit = NestList[Simplify[f[#]] &, 1/GoldenRatio, 1000];
Grid[{
{$Version, SpanFromLeft},
{"(Machine) Min:", N[Min[neworbit]]},
{"Min:", N[Min[neworbit], 2]},
{"Max:", N[Max[neworbit], 2]},
{Histogram[N[neworbit, 10], 20]},
{Histogram[neworbit, 10]}},
Alignment -> {Right, Center}]
]