Is it fair to say that most real-life distributions have finite variance? In most real scenarios I know the random variable X isn't unbounded. 
 A: If the data for a variate are directly recorded in some way (not defined as a ratio of two other variables which are the actual ones stored, for example), the recorded variate will pretty much have to have bounded variance, because we don't have a way of recording an infinite number of figures.
Even without that issue, almost all measured variates are actually bounded in some way even when we model them by distributions that are not. For example, a lognormal distribution is sometimes used for sizes of things (rocks or incomes, for example) and that distribution has no upper bound, but no rock on earth could be bigger than the earth and no income can be bigger than the entire economy of the world. 
However, variates that are derived from other variates may not be so clearly bounded, at least in the sense that there's a readily identifiable finite upper limit (in particular situations, there generally will be some limit, but it may be impossible to identify other than knowing it must be astronomically large)
A: The Cauchy distribution  https://en.wikipedia.org/wiki/Cauchy_distribution is unbounded and does not have finite moments (mean, variance, etc.) of any positive integer order. Note that the Cauchy Principal Value https://en.wikipedia.org/wiki/Cauchy_principal_value of the mean exists and is equal to the median, and if this is used, the variance = infinity.
Per the Wikipedia article "The Cauchy distribution is the distribution of the X-intercept of a ray issuing from (x_0,gamma) with a uniformly distributed angle. Its importance in physics is the result of it being the solution to the differential equation describing forced resonance. In mathematics, it is closely related to the Poisson kernel, which is the fundamental solution for the Laplace equation in the upper half-plane. In spectroscopy, it is the description of the shape of spectral lines which are subject to homogeneous broadening in which all atoms interact in the same way with the frequency range contained in the line shape."
Per http://www.scienpress.com/Upload/TMA/Vol%201_1_3.pdf , "Because of this special nature, some authors consider the Cauchy distribution as a pathological case. However, it can be postulated as a model for describing data that arise as n realizations of the ratio of two normal random variables. Other applications given in literature: [4] found that Cauchy distribution describes the distribution of velocity differences induced by different vortex elements. An application of the Cauchy distribution to study the polar and non-polar liquids in porous glasses is given in [5]. In [1] pointed out that the Cauchy distribution describes the distribution of hypocenters on focal spheres of earthquakes. It is shown in the paper [7] that the source of fluctuations in contact window dimensions is variation in contact resistivity, and the contact resistivity is distributed as a Cauchy random variable."
I think that some of these these qualify as "real-life" applications. It may be fair to say that most, but not all, real-life distributions have finite variance.
