foreach my $close_to_zero (1e-3,
                           1e-4,
                           1e-5,
                           1e-6,
                           1e-7,
                           1e-8,
                           1e-9,
                           1e-10,
                           1e-11,
                           1e-12,
                           1e-13,
                          ){
  print $close_to_zero**$close_to_zero,"\n";
}
[download]

0.993116048420934
0.999079389984462
0.999884877372469
0.999986184584876
0.999998388191734
0.99999981579321
0.999999979276734
0.999999997697415
0.999999999746716
0.999999999972369
0.999999999997007
[download]

If you approach zero from the negative side, it doesn't work. There are many other directions you can approach from, using complex numbers. From these directions the answer also seems to head toward a limit of 1.

 use Math::Complex;

my $j=sqrt(-1)/100;

foreach my $close_to_zero (-1e-3-$j*1e-3,
                           -1e-4-$j*1e-4,
                           -1e-5-$j*1e-5,
                           -1e-6-$j*1e-6,
                           -1e-7-$j*1e-7,
                           -1e-8-$j*1e-8,
                           -1e-9-$j*1e-9,
                           -1e-10-$j*1e-10,
                           -1e-11-$j*1e-11,
                           -1e-12-$j*1e-12,
                           -1e-13-$j*1e-13,
                          ){
  print $close_to_zero**$close_to_zero,"\n";
}
[download]

1.00689492851834+0.00322274943578533i
1.00091826683245+0.000322665621543586i
1.00011482165982+3.24709453669811e-05i
1.00001378423429+3.2697926634863e-06i
1.00000160867421+3.29277874043755e-07i
1.00000018389316+3.31579990412048e-08i
1.0000000206919+3.33882521438423e-09i
1.00000000229945+3.36185100395793e-10i
1.00000000025297+3.38487684801375e-11i
1.0000000000276+3.40790269818147e-12i
1.00000000000299+3.43092854902761e-13i
[download]

So my engineering view is that 0**0==1 most of the time, for large values of zero.

It should work perfectly the first time! - toma