... %.{$var}f doesn't do any trick except to produce [a] ... warning.

works for me in ... perl 5, version 20 ... Doesn't throw any warnings.

I'm surprised to read that it works in 5.20. I don't see any mention of a new format specifier in the latest sprintf docs. The best I can test in ATM is 5.14:

c:\@Work\Perl\monks>perl -wMstrict -le
"print 'perl version: ', $];
 ;;
 my $p = 30;
 my $n = 1.111111111111111111111111111111;
 ;;
 printf qq{%0.{$p}f}, $n;
"
perl version: 5.014004
Invalid conversion in printf: "%0.{" at -e line 1.
%0.{30}f
[download]

... I was surprised that the data changed ...

As to this, I'm sure that, having now read the docs, you realize that IEEE-754 floating point representation (or any other finite representation) is only an approximation that may sometimes happen to be exact, and that it handles base-2 relative-prime fractions no better than base-10 handles 1/3.