Is there an easy way to see associativity or non-associativity from an operation's table?

Have you seen Light's associativity test? According to Wikipedia, "Direct verification of the associativity of a binary operation specified by a Cayley table is cumbersome and tedious. Light's associativity test greatly simplifies the task."

If nothing else, the existence of Light's algorithm seems to rule out the possibility that anyone knows an easy way to do it just by looking at the original Cayley table.

Note also that, in general, one cannot do better than the obvious method of just checking all $n^3$ identities of the form $(a\ast b)\ast c = a\ast (b\ast c)$. This is because it is possible that the operation could be completely associative except for one bad triple $\langle a,b,c\rangle$. So any method that purports to do better than this must only be able to do so in limited circumstances.

Using the original $n\times n$ table seems bleak - this is essentially a problem of arity-dimension three, but the Cayley table only gives us two dimensions. However, Light's Associativity Test shows how to systematically reduce the problem of comparing $n$ pairs of Cayley tables. Note that the procedure can be greatly simplified by considering only operations derived from the underlying structure's generators.

First of all, let me make a personal reflection on this matter. Light's associativity test (as others have noted) provides a characterization, but (at least from my point of view) it is not really helpful. Indeed, I like to consider this difficulty to check whether a table is associative as the main reason why it is better to introduce associative operations (in particular groups) through presentations. Then, you trivially get associativity since your "object" is by definition a quotient of the free one.

Now, let me note that in the particular case that the operation is commutative (like the example you have written) then it is known an alternative method which is affordable to be done using a pencil. This (quite unknown in my opinion) method is due to S. KAMAL ABDALI and was introduced in his paper "Verification of Associativity of a Binary Operation" http://www.jstor.org/stable/3613856

I have never seen this method explained in a book, so it is worthwhile to take a look at this paper (in case you can go through the publisher firewall).