When I was very young, I was told infinity was the largest number, and I was shown how to write it: ∞. But what about infinity plus one? ∞ + 1 = ∞. So the number-line I envisioned started: 1, 2, 3, 4, 5... and at some point we hit infinity and it goes: ∞, ∞, ∞, ∞, ∞, ∞... So once you reach infinity, everything after that is infinity; which is why I was told it was the largest number (there is no bigger number to the right of infinity).
Shortly after learning about this, I became curious about what lied to the left of all those infinities. My mind focused on the first one, and shifted one step to the right, and then back... Then I had a sort of revelation. After shifting rapidly between the first two infinities, I had a vision of them shooting off to the left, and it hit me: If ∞ + 1 = ∞, then ∞ - 1 = ∞. When I discussed this idea my older brother insisted that I was wrong, but my father confirmed that I was right. I didn't understand how the infinities connected to the rest of the numbers on my number-line, but the discovery was exciting and mysterious. I was fully willing to accept that to the right there was an endless patch of infinities, while all the familiar numbers where on the left. To cross from one side to the other would require an infinite number of steps.
There was a book I grew up with called The Volume Library. This was an old book which belonged to my father. It was basically an attempt to merge every grade-school textbook you would need into one text of over two thousand pages of small print; and in here I found my introduction to Zeno's paradoxes.
The paradoxes of Zeno are important in the history of mathematics because they are related to the knotty problem of continuity, in many ways the central problem of mathematics. The best known of Zeno's paradoxes is that of Achilles and the Tortoise. Suppose that a tortoise is fleeing from Achilles, and that Achilles runs 10 paces while the tortoise runs 1. Suppose that Achilles, at the beginning of the pursuit, is 10 paces behind the tortoise. When Achilles has covered these 10 paces, the tortoise will be 1 pace ahead; when Achilles covers that distance, the tortoise will be 1/10 of a pace ahead; when Achilles has advanced this 1/10 of a pace, the tortoise will be 1/100 of a pace ahead. It is evident, therefore, said Zeno, that Achilles can never overtake the tortoise. Since we know that in real life Achilles does overtake the tortoise, it may seem that this cavil is of little consequence. But paradox or contradiction in formal logic is intolerable to the mathematician, for it destroys his whole science. Zeno must be answered not by an appeal to the facts of experience but by an appeal to logic. As a matter of fact, Zeno's paradox and others like it have never been answered to the satisfaction of all mathematicians and have lain at the bottom of many fiercely contested disputes in mathematics and philosophy. -from: The Volume Library
Now my way of thinking was visual, and my understanding of numbers was geometric. Viewing this through the same lens with which I saw those infinities, it lead me to an interesting conclusion: infinities can have a beginning and an end. Not only can a distance with a beginning and end be infinite, but all distances are infinite. When we mark a ruler, we may use different units of measure (like inches or centimeters). And between each full mark for an inch, we have a smaller mark for each 1/2 inch, then we have 1/4 and 1/8 inches; but we could identify these same lengths with whatever units we preferred.
So we could set a point and mark another point one inch to the right, or... we could just look at two points and use the distance between them as a unit for a custom measure. So if we kept looking at smaller measurements 1/16 inch, 1/32 inch... it leads us to 1/∞ inch. Instead of saying 1/∞, let's call this unit of measure an omega. So instead of saying 1/∞ inch, we can just say one omega; and we can say that an inch is infinity omegas long.
But we could also have half an omega, 1/4 an omega and so on. If we do the same with 1/∞ of an omega, we'll see that even the omega is infinitely long, as will be any unit we decide to use. There's a bit of a problem here though, we can't really tell one infinity from another with the notation we're using. As a child, I couldn't even explain things as well as I did here; so all my attempts to convince anyone that infinities could have a beginning and an end resulted in failure.
I've learned a bit since then though, so if you want to see a geometric figure with infinite proportions, here it is. Not only does the Koch curve have infinite perimeter, but the section between any two distinct points along the perimeter will also be infinite.