Examples of rational numbers with large denominators appearing unexpectedly. I am looking for examples of rational numbers with large denominators that pop up in questions in which there is a-priori no large numbers involved or no obvious reason why the denominator would be large. (for example $1/\Sigma(11)$  would rather clearly have a large denominator, where $\Sigma$ is the busy beaver function).
I'll start with an example to get it rolling.
We can consider this problem which asks for diofantine equations with only titanic solutions A diophantine equation with only "titanic" solutions. So we can obtain a rational with large denominator by simply taking $\frac{1}{n}$ where $n$ is a solution to one of those equations.
It would also be great if someone could give an example of a problem in which a certain constant that comes up looks like it may be irrational, but in the end turns out to be rational but with very large denominator.
I thought about going through the wikipedia page on mathematical constants to see if a rational one with large denominator appeared, but none did. Although it is possible this is just because if one was rational there would be no "need" for a special symbol for it.
Thank you and best regards.
 A: The Borwein integrals are a frequently discussed example where large denominators unexpectedly appear. One has
$$\int_0^\infty \frac{\sin(x)}{x} \ dx = \frac{\pi}{2}$$
$$\int_0^\infty \frac{\sin(x)}{x} \frac{\sin(x/3)}{x/3}\ dx = \frac{\pi}{2}$$
$$\int_0^\infty \frac{\sin(x)}{x} \frac{\sin(x/3)}{x/3}\frac{\sin(x/5)}{x/5}\ dx = \frac{\pi}{2}$$
and so on, until suddenly
$$\int_0^\infty \frac{\sin(x)}{x} \frac{\sin(x/3)}{x/3} \cdots \frac{\sin(x/15)}{x/15}\ dx =  \frac{467807924713440738696537864469}{935615849440640907310521750000}\pi.$$
A: Here is a rather natural example which I expect to blow out of the water any other response. Let us consider the set of fusible numbers, which is the smallest set containing $0$ and closed under the operation $(a,b)\mapsto\frac{a+b+1}{2}$ for $|a-b|<1$. This operation and its name originate from a classical puzzle of trying to measure 45 seconds using a pair of 1-minute long fuses, which amounts to showing $3/4$ is fusible.
Anyway, this set can be shown to be well-ordered, and hence for any $n\in\mathbb N$ it makes sense to ask for the smallest fusible number greater than $n$. It is not hard to show it will be of the form $n+\frac{1}{2^{m(n)}}$ for some $m(n)\in\mathbb N$. These values are small at first, with $m(0)=1,m(1)=3,m(2)=10$, but already $m(3)$ is gargantuan - for these familiar with Knuth's up-arrow notation, we have $m(3)>2\uparrow^9 16$. Hence the least fusible number larger than $3$ will constitute a rational with (in my opinion surprisingly) extremely large denominator, exceeding $2^{2\uparrow^9 16}$.
Of course as we increase $n$ these numbers get yet larger, with some bounds in terms of the fast-growing hierarchy proven in this survey.
A: Convergents of continued fraction expansions can have unexpectedly large denominators.
For example, we are all familiar with the 2nd convergent
$$\frac{22}{7} = [3;7] = 3 + \frac{1}{7}
$$
of the continued fraction expansion of $\pi$.
The 3rd convergent is
$$\frac{318}{106} = [3;7;15] = 3 + \frac{1}{7 + \frac{1}{15}} = 3 + \frac{15}{106}
$$
And the 4th convergent is
$$\frac{355}{113} = [3;7;15;1] = 3 + \frac{1}{6 + \frac{1}{15 + \frac{1}{1}}}
$$
But then the denominator of the 5th convergent jumps unexpectedly
$$\frac{103993}{33102} = [3;7;15;1;292] = 3 + \frac{1}{6 + \frac{1}{15 + \frac{1}{1 + \frac{1}{292}}}}
$$

The "unexpected" phenomenon that is occurring here is a sudden jump in the size of the denominator from one convergent to the next, correlated with a large term in the continued fraction expansion (in this case 292), correlated with a sudden shrinkage of the approximation error from one convergent to the next. Compare, for example,
$$\frac{355}{113} \approx 3.14159292
$$
which is an approximation of $\pi$ that is accurate to 6 decimal places, with the next approximation
$$\frac{103993}{33102} \approx 3.1415926530
$$
which is accurate to 9 decimal places.
