On my system, ∞ is inf, so I needed to lowercase the I.

Here's my shorter version:

#                 1         2         3
#        12345678901234567890123456789012345
perl -E 'say$_=3;/i/?die:say$_*=$_ while--$_' # old
perl -E 'say$_=3;say$_*=--$_ while$_&&!/i/'   # new
[download]

($q=q:Sq=~/;[c](.)(.)/;chr(-||-|5+lengthSq)`"S|oS2"`map{chr |+ord
}map{substrSq`S_+|`|}3E|-|`7**2-3:)=~y+S|`+$1,++print+eval$q,q,a,
[download]

Playing with your proposal a bit leads to

$_=3;say,$_*=--$_ while!/i/
[download]

Without using a bigint module, you can't generate much of the series. I suspect correct Perl 6 solutions would be really easy.

use bigint; my $f=3; for (1..12) { $f = ($f-1)**2 if $_>1; say $f }
# Faster: replace bigint with Math::GMP qw/:constant/;
[download]

I admit I find code golf to be tiresome, and prefer readable and efficient solutions.

I suspect correct Perl 6 solutions would be really easy.

Yes, indeed, really easy in Perl 6, except that you can't seem to get to infinity. ;-) So there is a need for some other way of stopping the calculation than just looking for /inf/ or the like.

In fact, it is so easy that I just needed to copy and paste your code and run it with Perl 6/Rakudo:

$ perl6 -e 'my $f=3; for (1..16) { $f = ($f-1)**2 if $_>1; say $f }'
3
4
9
64
3969
15745024
247905749270529
61457260521381894004129398784
...
[download]

The full output for those interested with big nums (adding an extra newline between the numbers for a better clarity of output) is given below:
Read more... (22 kB)
The last number above has more than 7300 digits. The whole one-liner script executes in less than 1 second, but it starts to show a noticeable slow down when getting above 16.

Trolling season!

perl -E 'say(eval),$,.=$#$,$,x=eval,redo,for$,="+3"'