I do not recall the beginning of this conversation, but I do recall that we were eating pizza at the dinner table when Tabitha anticipated my turn in the dicussion.
Tabitha (6 years old): I know what you’re going to say, Daddy. “Counting never ends.”
Me: I suppose that sounds like something I would say, yes.
T: What’s the biggest number, though? Googolplex?
Quick tutorial. A “googol”—spelled that way—refers to this number: 
It is, of course, a very big number. Far too big to be practical in any meaningful sense. The very idea of such a large number having a name is fascinating to children. Most children (in my experience) encounter one googol in their social interactions with other children. The googol does not appear in the Common Core State Standards.
A “googolplex” is 
You may Google googol for lots of interesting characterizations of how extremely silly this very large number is.
For example, you will not live for one googol seconds.
Indeed, the universe has not existed for one googol seconds (not even by the greatest estimates of its age—not even close).
You get the idea.
Me: Well, like you said I would say, counting never ends, so no googolplex is not the biggest number.
T: If you counted by 10,000 could you ever get to googolplex in your life?
Me: No.
T: If you counted by 11,000?
Me: No.
T: 12,000? 13,000?
Me: No. Even if you counted by googol, you couldn’t get to googolplex in your lifetime.
T: Well, what if you counted by googolplex?
Me: Well sure. It would the start of your count, wouldn’t it?
She decides to demonstrate this (Side note, we have been counting by various numbers of late).
T: Googolplex.
She smiles broadly, congratulating herself for successfully counting to what she has perceived to be the largest number.
We discuss further the existence of a largest number. Then Tabitha makes a claim that takes us in a different direction.
T: Eventually, numbers just go back to the beginning.
Me: So if you keep counting, you get to zero?
T: No.
Me: One?
T: No, Daddy! Don’t you remember there are numbers before zero?
So what do we learn?
Big numbers are fun. Boy howdy are big numbers fun. Children love to talk about the biggest number, and whether one exists. There is all kinds of lovely thinking going on when they ask these kinds of questions.
Talking about big numbers often leads to talking about infinity. If there is no biggest number, it is because numbers go on forever. The only thing Tabitha has experience with that goes on forever is a loop. She drew on that loop metaphor in imagining that numbers go back to the beginning eventually.
Starting the conversation
Listen for the biggest number talk. It often surfaces when children are comparing their athletic prowess (I can jump 2 sidewalk squares! I can jump 100 sidewalk squares! Pretty soon, someone is claiming to be able to jump googol or infinity sidewalk squares.)
When it surfaces, support it. Play and explore with your child. Answer questions. Ask questions. Talk about it and have fun. Look stuff up together when the questions go past your own knowledge. Shoot me a question here at Talking Math with Your Kids if I can answer any of those for you.