First, with the irrelevant aside: vaccines aren't made of pure viruses. They're mostly filler.

Second, here is what I recall about Dr Offit's chain of reasoning.

  1. You need some amount of virus particles to provoke the
     immune reaction, call it 1 million.  (I'm too lazy to
     try to google the numbers myself.)
  2. You need some amount of immune cells of the various
     types to carry out the immune reaction which results in
     immunity, call it 10 million.
  3. You have a total of 1 trillion of these immune cells in
     your body.  (Once again, I'm making up the exact
     numbers.)
  4. So therefore, the capacity of the immune system is
     sufficient to deal with
     1 trillion / 10 million = 100,000 vaccines at once.

Third, and most importantly, of course 10,000 vaccines at once is hyperbole, but it's making a very important point. The question wasn't "Should we give our childrens 1 vaccine or 10,000 vaccines?", it was a debate about whether a three-vaccine (or was it seven?) combo "stressed" the immune system too much.

Dr Offit's point wasn't "Nah, let's give them 10,000 vaccines!" It was that when you go from 1 vaccine to 7 vaccines, you're increasing the load on your immune system from 0.01% to 0.07% and we've all got better things to worry about than that.

That's not a good example because you can easily combine multiple vaccines into a single injection.

The person who made the original claim does not post on HN, but the person who thinks that it is "certainly" unsafe does post here. Use of the word "certainly" implies that the answer is so obvious that we should all know the reasons why. Well, I don't. So I asked.