Intercept term in logistic regression Suppose we have the following logistic regression model:
$$\text{logit}(p) = \beta_0+\beta_{1}x_{1} + \beta_{2}x_{2}$$
Is $\beta_0$ the odds of the event when $x_1 = 0$ and $x_2=0$? In other words, it is the odds of the event when $x_1$ and $x_2$ are at the lowest levels (even if this is not 0)? For example, if $x_1$ and $x_2$ take only the values $2$ and $3$ then we cannot set them to 0.
 A: There also might be a case when $x_1$ and $x_2$ can not be equal to $0$ at the same time. In this case $\beta_0$ does not have clear interpretation. 
Otherwise $\beta_0$ has an interpretation - it shifts the log of the odds to its factual value, if no one variable can not do this. 
A: $\beta_0$ is not the odds of the event when $x_1 = x_2 = 0$, it is the log of the odds.  In addition, it is the log odds only when $x_1 = x_2 = 0$, not when they are at their lowest non-zero values.  
A: I suggest to look at it a different way ... 
In logistic regression we predict some binary class {0 or 1} by calculating the probability of likelihood, which is the actual output of $\text{logit}(p)$. 
This, of course, is assuming that the log-odds can reasonably be described by a linear function -- e.g., $\beta_0 + \beta_1x_1 + \beta_2x_2+ \dotsm   $
... This is a big assumption, and only sometimes holds true. If those $x_i$ components don't have independent, proportional influence on the log-odds, then best to choose another statistical framework.  I.e., the log-odds is made up of some fixed component $\beta_0$, and increased incrementally by each successive term, $\beta_i x_i$. 
In short, the $\beta_0$ value is the "fixed component" of that component-wise method to describe the log-odds of whatever event/condition you are trying to predict. Also remember that a regression is ultimately describing some conditional average, given a set of $x_i$ values. None of those things require that $x_i$-values be 0 in your data or even possible in reality. The $\beta_0$ simply shifts that linear expression up or down so that the variable components are most accurate. 
Maybe I said the same thing in slightly different mindset, but I hope this helps ...
