Is there a mathematical notation for "repeat $X$ until $Y$"?
The first one:
"Sum of reciprocals of natural numbers (starting with 1) until sum is above 42"
is fairly straightforward. First, the number where it goes above 42 is:
$$X = \min \left\{ x \in \mathbb{N} \,\left\vert\, \sum_{n=1}^x \frac{1}{n} >42 \right.\right\}$$
And so if you want to know what the sum is:
$$\sum_{n=1}^X \frac{1}{n}$$
Or, as one expression:
$$\sum_{n=1}^{\min \left\{x \in \mathbb{N} \,\left\vert\, \sum_{n=1}^x \frac{1}{n} >42 \right.\right\} } \frac{1}{n}$$
For the "Start with $2$ and keep squaring while result is less than 15"
you indeed need to talk about iterations of numbers. For this, we could define a recursive function. So, let $f$ be a funtion from natural numbers to natural numbers defined by:
$f(1)=2$
$f(x+1)=f(x)^2$ (for $x>1$)
And now the result we are looking for is:
$$\max \{ f(x) \mid x \in \mathbb{N}, f(x)<15 \}$$
I think a lot of times in mathematics, we are less interested in the procedure and more interested in the result. So, why do you need to do $X$ until $Y$? Presumably it is to compute some value. Often in a mathematical context, it is sufficient to just assert (or prove) the existence of such a value without regard to how it's computed.
Consider your example: "Sum the reciprocals of natural numbers until the total is above 42." As someone not interested in futility, I would ask why I should do this. Perhaps you are interested in the first such total that is above 42? If it is obvious that such a quantity exists, in mathematical literature we might instead say "Let $x_n$ denote the sum of the reciprocals of the first $n$ natural numbers, and let $x$ denote the least element of the set $\{x_n \; |\; x_n > 42\}$". Very often I am not interested in how one might go about computing $x$.
I think there are a few ways to do it, but probably the most natural is to use set-builder notation.
You can take the sequence $x_{n+1}=x_n^2$ which is a recursively defined sequence, and you could take $$\{x_n \mid x_n<15\}$$ to capture the repitition. and if you just want the last one, $$\mathrm{max}\{x_n \mid x_n<15\}.$$
This kind of recursion approach along with set-builder notation, or "cases," is adopted in Haskell.
However, mathematics is usually communicated with common language. Just saying, "repeatedly square $x$ while $x<15$" would probably do the trick for the recursion, or maybe "let $a_n$ be the largest member of the sequence $a_{n+1}=a_n^2$ so that $a_n<15$." I think the computer science notation is out of necessity for a computer to understand you, which is not a requirement for communicating math.