I have heard that Rav Scheinberg wore many garments with ציציות on them at a time. . .
Regarding Rav Scheinberg's daughter who was once ill,he did not take on extra tzitzis, that is only why he doesn't speak on Shabbos. He has not publicized the reason for his wearing multiple pairs of tzitzis, though he has stated that every pair is a mitzvah. He makes one bracha, if any, since the talis gadol can patur the bracha on tzitzis. He dons bundles of 18 at at time.
Rav Sheinberg originally put on 18 as a neder if his daughter recovered from an illness many years ago. He has added more since.
I assume he only makes 1 bracha just like "we" only make 1 on both our talis gadol and katan.
I was told that he wore multiple tzitzis garments to be yotzei all the shittos (e.g. Yemenite-Rambam tying, Sephardic tying, etc.), but this doesn't make much sense to me since I figure 5 or 10 or 20 should be plenty.
Moreinu HaGaon HaRav HaTzaddik Chaim Pinchas Scheinberg, zatzal, did not (contrary to public belief) wear multiple sets of tzitzis due to a desire to be yotzi according to all views. All of his tzitzis were the same.
He did this because (1) each beged is a mitzvah, each second that one wears it and (2) b/c there are non-religious folks who don't do the mitzvah, so he wanted to "make up" for them.
There was a time when he wore almost 300 sets of tzitzis. He had skin problems because of this because it is very hot in Israel and his apartment (at one point) had no AC.
Source: A Gadol in Our Midst