Questions that ask for "intuitive" reasons are admittedly subjective, but I suspect some people will find this interesting.
Some time ago, I was struck by the coincidence that the Euler-Mascheroni constant $\gamma$ is close to the square root of $1/3$. (Their numerical values are about $0.57722$ and $0.57735$ respectively.)
Is there any informal or intuitive reason for this? For example, can we find a series converging to $\gamma$ and a series converging to $\sqrt{1/3}$ whose terms are close to each other?
An example of the kind of argument I have in mind can be found in Noam Elkies' list of one-page papers, where he gives a "reason" that $\pi$ is slightly less than $\sqrt{10}$. (Essentially, take $\sum\frac1{n^2}=\pi^2/6$ as known, and then bound that series above by a telescoping series whose sum is $10/6$.)
There are lots of ways to get series that converge quickly to $\sqrt{1/3}$. For example, taking advantage of the fact that $(4/7)^2\approx1/3$, we can write $$ \sqrt{\frac{1}{3}}=(\frac{16}{48})^{1/2} =(\frac{16}{49}\cdot\frac{49}{48})^{1/2}=\frac{4}{7}(1+\frac{1}{48})^{1/2} $$ which we can expand as a binomial series, so $\frac{4}{7}\cdot\frac{97}{96}$ is an example of a good approximation to $\sqrt{1/3}$. Can we also get good approximations to $\gamma$ by using series that converge quickly, and can we find the "right" pair of series that shows "why" $\gamma$ is slightly less than $\sqrt{1/3}$?
Another type of argument that's out there, showing "why" $\pi$ is slightly less than $22/7$, involves a particular definite integral of a "small" function that evaluates to $\frac{22}{7}-\pi$. So, are there any definite integrals of "small" functions that evaluate to $\sqrt{\frac13}-\gamma$ or $\frac13-\gamma^2$?
I have an interesting approach. The Shafer-Fink inequality and its generalization allow to devise algebraic approximations of the arctangent function with an arbitrary uniform accuracy. By a change of variable, the same holds for the hyperbolic arctangent function over the interval $(0,1)$ and for the logarithm function over the same interval. For instance,
$$ \forall x\in(0,1),\qquad \log(x)\approx\frac{90(x-1)}{7(x+1)+12\sqrt{x}+32\sqrt{2x+(x+1)\sqrt{x}}}\tag{A} $$ and $\approx$ holds as a $\leq$, actually. We have $$ \gamma = \int_{0}^{1}-\log(-\log x)\,dx \tag{B}$$ hence: $$ \gamma\leq 1-\log(90)+\int_{0}^{1}\log\left[7(x+1)+12\sqrt{x}+32\sqrt{2x+(x+1)\sqrt{x}}\right]\,dx\tag{C}$$ where the RHS of $(C)$ just depends on the (logarithms of the) roots of $7 + 32 x + 12 x^2 + 32 x^3 + 7 x^4$, which is a quartic and palindromic polynomial.
The numerical approximation produced by $(C)$ allows to state: $$ \gamma < 0.5773534 < \frac{\pi}{2e}.\tag{D}$$ Actually $(A)$ is not powerful enough to prove $\gamma<\frac{1}{\sqrt{3}}$, but we can achieve that too by replacing $(A)$ with the higher-order (generalized) Shafer-Fink approximation.