I'm inclined to think the bc-calculated value is accurate

Yes, and the mpfr library agrees:

C:\>perl -MMath::MPFR=":mpfr" -le "Rmpfr_set_default_prec(1000000); $x
+=Math::MPFR->new('1.000001'); $x **= 60000000; $x *= 10; Rmpfr_out_st
+r($x, 10,43,MPFR_RNDN);
1.141973130130727445029596475971653735685612e27
[download]

That's calculated with a million bits of precision - and gives the same result as when calculated with a thousand bits of precision.
So I think we can rest assured that that is the correct figure for 43 decimal digits.

Cheers,
Rob