Pensieve header: How wasteful are continued fractions information-theoretically, as means to store integers?

r=RandomReal[WorkingPrecision->40]

RealDigits[r][[1]]//Length

cf=ContinuedFraction[r]

Length/@IntegerDigits/@cf

N[FromContinuedFraction[cf],50]

Total[Length/@IntegerDigits/@cf]

N@Mean[Table[
r=RandomReal[WorkingPrecision->100000];
cf=ContinuedFraction[r];
Total[Length/@Rest[IntegerDigits/@cf]],
{1000}
]/100000]

Plot[ContinuedFraction[x][[2]],{x,0,1}, PlotPoints -> 1000]

N@Sum[ ((10^(1-n))-(10^(-n)))(n-Log[10,(10^(1-n))-(10^(-n))]),{n,1,200}]

N@Sum[ ((10^(1-n))-(10^(-n)))n,{n,1,200}]

ContinuedFraction[0.010000001]