The title says it all: what is the closest proven lower-bound on the prime counting function?
Note that I'm not limiting this to continuous functions. If a lower-bound on the prime-counting function exists over any infinite domain and is consistently closer than a continuous contemporary, I'm interested in the discrete function.
The closest lower bound on $\pi(x)$ depends upon the minimum value of $x$ that you are talking about because, as $x$ increases, the best known estimate for the lower bound becomes sharper and sharper. Example
For $x \ge 11$ we have
$$ \pi(x) \ge \frac{x}{\ln x}. $$
For $x \ge 5393$ we have
$$ \pi(x) \ge \frac{x}{\ln x - 1}. $$
For $x \ge 88783$ we have
$$ \pi(x) \ge \frac{x}{\ln x} \Big(1 + \frac{1}{\ln x} + \frac{2}{\ln^2 x}\Big). $$
In general for any given $x$ it is possible to find the lower bound on $\pi(x)$ and so very few number theorist are interested in finding such explicit bounds because there is no end to it and such bounds do not provide as much information as an asymptotic formula anyways.