#Perl 5.10+, 40 37 bytes
$_=q{say"\$_=q{$_};eval"while 1};eval or (also 37 bytes)
$_=q{{say"\$_=q{$_};eval";redo}};eval Invoke with the -M5.010 or -E command line flag, e.g.
$ perl -E '$_=q{say"\$_=q{$_};eval"while 1};eval' $_=q{say"\$_=q{$_};eval"while 1};eval $_=q{say"\$_=q{$_};eval"while 1};eval $_=q{say"\$_=q{$_};eval"while 1};eval ... Thanks to Ilmari Karonen for shaving off 3 bytes from my original solution, which was:
eval while$_=q{say"eval while\$_=q{$_}"} This, as well as the shorter 37-byte solutions above, are all simple variations of the following quine, which I first saw in one of Ilmari's other posts:
$_=q{say"\$_=q{$_};eval"};eval Since all I added in my original solution was a while loop, he really deserves most of the credit. :-)