The heaviest tailed smooth normalizable continuous distributions that I am familiar with are those with [fat][1] power-law tails $\frac{1}{x^{1+\alpha}}$, e.g. a [Pareto][2] with $\alpha\rightarrow 0^+$ or a [Student's t][3] with $\nu\rightarrow 0^+$, but are there distributions with heavier tails?  I am curious about what is the worst case possible for a distribution that decreases monotonically away from a peak positive value towards a minimum of 0.

I think that the heaviest possible normalizable heavy tails are indeed those asymptotic to $\frac{k}{x}$ as $x\rightarrow\infty$ (where $k$ is some constant), but have not been able to prove it to my satisfaction nor find a clear statement of this in the literature. I wonder if my belief is either obvious to experts or wrong.

A couple of notes: 

 - A function $f(x)$ is heavier-tailed than $g(x)$ for $x>0$ if there exists some finite $x_0$ such that for all $x>x_0$, $\int_{x_0}^\infty f(x)\,dx > \int_{x_0}^\infty g(x)\, dx$. (As discussed in the answer to: https://stats.stackexchange.com/questions/86429/which-has-the-heavier-tail-lognormal-or-gamma)
 - It does not matter that the distribution has no finite moments, just that it has a finite median and integrates to 1 over the range  $[0,\infty]$ (one-sided) or $[-\infty,\infty]$ (two-sided).


  [1]: https://en.wikipedia.org/wiki/Fat-tailed_distribution
  [2]: https://en.wikipedia.org/wiki/Pareto_distribution
  [3]: https://en.wikipedia.org/wiki/Student%27s_t-distribution
  [4]: https://en.wikipedia.org/wiki/Dirac_delta_function