Why is $15-\sqrt{15}-\sqrt{15-\sqrt{15}}-\sqrt{15-\sqrt{15}-\sqrt{15-\sqrt{15}}}$ so close to $5$? Basically I started with the number $15$. Then I subtracted its square root to get roughly $11.127$.
Subtracting the square root of that returned roughly $7.791$, and finally after taking the square root of that, I got roughly $5$. 
According to Google, it is $5.00000861488$. It is so close to $5$ that I though it could use further investigation, but so far I haven’t found anything like it.

I don't think there is anything fundamental about this.  Here's a plot of the real part for repeated applications of the process:

Mathematica:
iter = Re[N[NestList[(# - Sqrt[#]) &, 15, 15]]];

ListPlot[iter, Joined -> True,
 PlotRange -> {-2, 15}]

