I guess it depends on what sense of infinity you’re trying to define. What is your motivation for trying to explain it to a lay person? All these definitions are good, although most are poetic. A mathematical definition of infinity (technically countable infinity, or aleph 0, the smallest infinite cardinal) would be a quantity that is bigger than every natural number.
There are also lots of weird things that you can do with infinity.
For example, there are the same number of even numbers as whole numbers, and even the same number of whole numbers as rational numbers. (A rational number is basically any fraction of whole numbers).
But there are infinitely more real numbers than rational numbers. If you take a number line and throw a dart at it, there is a 0% probability that you will hit a rational number; you will hit an irrational number every single time. This is suprising because the rational numbers are “dense,” which means that if you give me any real number, and any small number r, no matter how small, I can always find an infinite number of rational numbers that are less than r away from your starting number. Mathematicians describe the fact that there are infinitely more real numbers than rational numbers by saying that the real numbers have larger cardinality than the whole numbers. We call the number of real numbers “the cardinality of the continuum” and denote it aleph 1. There are also even bigger infinities, called aleph 2, aleph 3, etcetera. Even this sequence goes up to infinity.
Another fun problem is the infinite hotel. Say you have a hotel with an infinite number of rooms, which are numbered 1,2,3,4,5,etcetera. They are all full. However, the guests in these rooms are very accomadating of the management, and willing to move into a new room whenever asked. The manager has an intercom that talks to all the rooms and gives them instructions on how to move rooms.
So the hotel is full. You show up and want to check in. “No problem,” says the manager, you can stay in room 1. He then goes on the intercom, and tells every guest to move into the next highest numbered room. Room 1 is now empty, and you move in. This works because there is no absolute largest number.
This will even work if an infinitely large bus shows up with infinitely many people on it, and they all want to check in. The manager tells every guest to move into the room whose number is twice their current room number. Now all of the odd numbered rooms are empty, and there are infinitely many odd numbered rooms, so you can accommodate the whole bus.
