Non-Compact Domain/Range Machine Learning What are some machine learning methods that can be applied to non-compact input domain and output range?

Neural Networks is clearly not one of them as they well approximate in an interval.

I have a problem at hand where I need to identify a function with domain the entire real line. The good thing is I know the asymptotic behavior, which is a finite point. 

Unfortunately common ML methods diverge immediately outside of training bounds and cannot be hoped to have any asymptotic stability. Polynomials were one idea that came to my mind, but they also diverge asymptotically.
Notice that the limitation of finite domains and ranges stems from the finite nature of hardware and thus finite representation of numbers. Therefore, a more promising approach is remodeling the domain, rather than using an "infinite" algorithm.

You can apply logarithmic scaling to your domain and range. This will get you off from the open bounds while still preserving order and continuity, which is important fur numerical ML algorithms.

The log function is the basic building block that you can use for your transformation. You can scale it to fit your ranges and this would solve your positive part of the range. You can do the same for the negative, even using  different coefficients if this suits you.