The word *any*, when used in a negation, is not a double negative but a **Negative Polarity Item**, which is defined by the paper [Definite Descriptions and Negative Polarity][1] as words that “seem happy under negation and are sometimes unhappy without negation”.

When you have a double negative, “standard” English dictates that you need to use one or the other but not both:

 - I can’t get satisfaction. 
 - I can get no satisfaction. 

In contrast, you don’t get a grammatical sentence with *any*:

 - I can get any satisfaction. 

This would be grammatical as part of a longer sentence but note then how it doesn’t mean “no satisfaction”. Note also that there’s another sense of *any* (“free choice *any*”) with a completely different meaning that would be grammatical in similar sentences because it’s not a NPI (“I can get any TV” meaning that no TVs are out of your budget). 


  [1]: http://danielrothschild.com/npidd.pdf