Yes, but it is infinite if you never stop doing it.
Unlike something like 1+.1+.01+.001+.0001…
No matter how long you do it, it will never reach 2, much less infinity.
It’s like people saying that pi is infinite. Well, no, it’s less than 4 (and also less than 3.2, and less than 3.15, and less than 3.142, and…). One common way of representing numbers would lead to an infinitely-long representation of it, but the even though the representation is infinite, the number itself is still finite.
if you have a geometric sequence like r^n and r>1 that supposedly converges to some finite limit L, and say e < L(r-1)/r, then r(L-e) is already greater than L.
ETA no need to worry about representing any numbers. If a number (in your case the limit of some sequence) is less than 2 then it is not infinite, doesn’t matter what the exact value is.
It’s not circular, it’s definitional. “…” is mathematical speak for “continue to infinity”.
Fine.
I am still not seeing where measuring a coastline extends into infinity.
I agree with that. For a coastline the uncertainties from how we define what is part of the coast and what isn’t becomes unmanageable way before we even have to worry about how to measure distance along a molecule. Similar issues apply to all physical objects.
But the coastline paradox, at least as stated by Wikipedia, doesn’t actually say coastlines tends toward infinity, just that the length of a coastline isn’t well-defined and that choosing a shorter ruler will increase the length. This is true down to rulers so short the practical exercise of measuring becomes impractical.
Mathematical fractals on the other hand can tend towards infinite length inside a finite area.
depends on how far in you can zoom into the coastline
Obviously, in the “real world” almost nothing is infinite. However, that you do have this problem leads to the question from a mathematical perspective, where you can use infinites.
Should you ever decide to measure some coastlines, I suggest the use of satellite imagery and image-processing software over rulers, as a practical matter 
Just to be clear, to measure the length of a coastline using satellite imagery and imaging-processing software, you are still choosing a ruler in the sense I intended here. It becomes a bit more complicated though because it involves the resolution of the image, the ability of the processing software to tell water from shore, the algorithms for deciding what the length of the jagged intersection between water pixels and shore pixels is, etc.
As a practical matter, take a look at Wikipedia’s List of countries by length of coastline. They use data from two different sources, both reputable, and both presumably trying to make the best measurement they can, but the numbers are wildly different. For instance, the World Factbook says that the US has 19,924 km of coastline, but the World Resources Institute says it’s 133,312 km. And Ireland, to use Velocity’s example, goes from 1,448 to 6,437 km.
I’m sure that neither of these sources is counting around individual pebbles on the beach, or grains of sand, much less molecules or atoms. But even using whatever reasonable standards they’re applying, there are still major variations.
Every time you add triangles to the border, you are enclosing more space, so the area has to get bigger. Conceptually I don’t see a way around this. The area seems to be getting infinitely bigger, but not infinitely big, whereas the perimeter is getting infinitely bigger (or longer) and infinitely big.
The sticking point for me with the snowflake is that, while you can say the perimeter is getting more convoluted, it is only getting convoluted in one direction- outward. That is why you are continually adding area to the inside. If the border was also convoluting inward- as it would with a coastline- then with every iteration to the perimeter you’d be adding and subtracting from the area. Then I could easily see how the perimeter can get infinitely longer while the area does not get infinitely bigger.
FTR: I never meant to suggest someone measure a coastline at the molecular (or atomic or Planck) level. I was only trying to note limits on how small we can make our measuring stick since that seems central to the discussion.
As long as you keep iterating, it will keep getting bigger, but every time, it will get bigger by a smaller and smaller amount. It will approach a particular size, but never reach nor pass that size.
As has been said, you could draw a circle around it, and it will never escape that circle, meaning that it could never have an area greater than that circle.
I think your sticking point is the term “infinitely bigger”. Your own example is a good one. 1.2 is clearly a finite number. You can add smaller increments to 1 an infinite number of times but it never gets to 1.2, it is not getting “infinitely bigger”.
In the snowflake example, the area never goes beyond an upper boundary, but the perimeter has no boundary, it does get infinitely bigger.
Does it?
Each iteration is smaller than the one before it. I can see you get an infinite sequence but it is not getting infinitely bigger. Using your example, you infinitely approach 1.2 but you never get there.
You are not getting infinitely bigger. You are just getting infinitely closer at smaller and smaller increments.
Put another way, Norway’s coastline is not infinitely long (obviously). You are just getting infinitely closer to a set, finite value (whatever that is and I know that we cannot define that value perfectly).
If you give an initial perimeter for the 0-iteration snowflake and any goal length for the perimeter, no matter how large, the number of iterations to reach that length can be calculated. The perimeter genuinely tends towards infinity.
The increments are not getting smaller. They are actually increasing exponentially.
(4/3)^n, where n is the number of increments.
I think you are saying you get more increments but each increment is smaller than the last.
So…?
It remains you are forever chasing something and never quite getting there.
No, you are getting more increments, and each is larger than the last.
Ok…you lost me (my problem, not yours).