> "an arbitrarily large number of objects that can all touch one another"

I think one must be careful to define "object" and "touch." Otherwise,
all these $n$ line segments (objects) intersect (touch) at the origin:

     [![enter image description here][1]][1]

But this example violates the spirit of the OP's questions.

 


  [1]: https://i.sstatic.net/squJN.jpg