jmc

Looking at the definition of $a♣b$, we see that $a♣b=\frac{2a}{b}\cdot \frac{b}{a}=\frac{2a\cdot b}{b\cdot a}=\frac{2ab}{ab}.$ Both the numerator and denominator share a common factor of ab, so $a♣b=\frac{2\cancel{ab}}{\cancel{ab}}=2.$ Thus, regardless of what a and b are (as long as neither are zero), $a♣b$ will always be 2. Looking at the given expression, a and b are never zero. Thus whatever the values of a and b, the expression will always evaluate to 2. Thus, the expression simplifies to $(5♣(3♣6))♣1=(5♣2)♣1=2♣1=\boxed{2}.$