I personally believe that

• a "better" s-based solution was already posted by jwkrahn, in particular:
• there's no need to match a 1 at the end of the string, and avoiding to do so could generalize the code to spit out the sequence to begin with zero 1's - not that anybody requested it, but it's such a tiny violation of YAGNI for both a simpler syntax and added flexibility that I would only avoid it if it were expressedly ruled out;
• we recommend not to use $a and $b as general purpose variables all the time, which would be especially important here, given that the OP is clearly a newbie;
• this thread is clearly facetious in many ways, but as a general programming rule, it's awkward and error prone that you somehow unnecessarily duplicate information, holding it in two variables where one would suffice: namely in $a as the length of $a itself and in $b as an integer, whereas the two quantities are fundamentally the same. Once you use only one variable, perhaps taking advantage of $_ to use smart defaults, and switching to the while statement modifier, your code could be morphed into:

perl -le 's/$/1/, print while 100 > length'
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Incidentally, now that I notice, the anchor is not necessary at all. If it were not for a mandatory /e modifier, this could make for a golfed solution just as good as JavaFan's

perl -E'{s//say/e,redo}'
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but that's not the case...