What is the smallest unknown natural number? There are several unknown numbers in mathematics, such as optimal constants in some inequalities.
Often it is enough to some estimates for these numbers from above and below, but finding the exact values is also interesting.
There are situations where such unknown numbers are necessarily natural numbers, for example in Ramsey theory.
For example, we know that there is a smallest integer $n$ such that any graph with $n$ vertices contains a complete or an independent subgraph of 10 vertices, but we don't know the exact value of $n$.

What kinds of unknown small (less than 100, say) integers are there?
What are the smallest unknown constants which are known to be integers?
Or, more rigorously, what is the smallest upper bound for an unknown but definable number that is known to be an integer?

I know that asking for the smallest unknown integer is ill-defined since we do not know the exact values.
The more rigorous version of the question is well-posed, but I do not want to keep anyone from offering interesting examples even if they are clearly not going to win the race for the lowest upper bound.

An answer should contain a definition of an integer quantity (or a family of them) and known lower and upper bounds (both of which should be integers, not infinite).
Conjectures about the actual value are also welcome.
I have given one example below to give an idea of what I'm looking for.
The smallest infinitely often occurring prime gap, or

$$\liminf_{n\to\infty}\; (p_{n+1} - p_n)$$

is unknown as of now. Most likely, it is $2$, but the twin prime conjecture has not yet been settled.

Due to the work of Yitang Zhang and subsequent improvements by others (notably James Maynard and Terence Tao), we know that some prime gaps occur infinitely often. Zhang proved that gaps not larger than 70 million occur infinitely often, and the improvements lowered the bound to $246$ (perhaps there have been recent further improvements I'm not aware of).
The number $\pi$ has a very significant history and could be the ideal object of many problems about the first position in its decimal expansion in which a given property occurs:

In what position the following property occurs first?

1) two consecutive twin numbers of two digits (six possible couples)

2) the number  5 appears 5 consecutive times

3) the first square of ten digits

4) Nothing is known about which digit appears countless times however it is clear that at least one digit does.  What is this digit?

Many other similar questions can be formulated. Note that, excepting 4), it is not obvious that there are corresponding answers but it is obvious also that their probability of existence is nonzero. A way to be sure of the existence of an answer is as follows: 

The famous BBP Formula

            

gives the value of any position in the expansion of $\pi$ so one can looking for some interesting property in an advanced position and ask about the minimal position at which this occurs (it can be the discovered position, of course).