"Why do I always get 1 when I keep hitting the square root button on my calculator?"

Here is how I had justified it to myself when I was a kid (I was convinced :-))

If $x > 1$ then $\sqrt x > 1$ and $x > \sqrt x$.

So we keep reducing the number while still being $> 1$.

Also, we cannot end up at a number $>1$ as then taking the square root would reduce it.

If we do end up at $1$, we stay there. Since the calculator has a limited precision, we end up at $1$, and pretty quickly.

Of course, the assumption here is that we do end up somewhere :-) (Which seemed justified by the fact that the first non-zero digit after the decimal point always seemed to reduce)

Make a picture of the usual spiral converging to the fixed point, in the style of

Where does this little obsession aparent on this site with 'layman' explanations? :P

NB. The picture comes from the Wikipedia page on fixed points.

Later. To make the picture it is better to use the cosine (so you do not even have to make the picture because that is what's in the wikipedia page) mainly because the iteration is seen more clearly: for the square root, the sequence converges boringly and fast to be interesting.

PS. You can draw this kind of pictures, assuming you have acces to Mathematica, with the following code:

f[t_] := Cos[t];
g = Plot[{f[t], t}, {t, -\[Pi]/2, \[Pi]/2}, 
  AspectRatio -> 1, PlotStyle -> {Thick}, Epilog ->
   Module[{pts = NestList[f, .1, 10]},
    {
     Thin, PointSize[0.015],  
     Line@ Flatten[Map[{#, {#[[2]], #[[2]]}} &, Partition[pts, 2, 1], 1], 1],
     Red, Point[Partition[pts, 2, 1]]
     }
   ]
]    

If they know about logarithms, try this:

• taking repeated square roots of a positive real number is the same as repeatedly dividing its logarithm by 2,
• repeatedly dividing something by 2 gets you to 0, and
• the exponential of 0 is 1.