Why is belonging not transitive? From Halmos's Naive Set Theory, section 1:

Observe, along the same lines, that inclusion is transitive, whereas belonging is not. Everyday examples, involving, for instance, super-organizations whose members are organizations, will readily occur to the interested reader.

Belonging seems transitive. Can someone explain?

Based on your response to other answers, your question seems to be "Why do we define the belonging relation ($\in$) in a way that cares about the level of nesting of sets?", i.e. why do we say that $a \notin \{\{a\}\}$?
We could define a relation that ignores nesting, a sort of recursive belonging that works the way you seem to want belonging to work (with $a {\tt\  recursively-belongs-to\ } \{\{a\}\}$), so why don't we use that as the definition of $\in$?
One reason is that we want to be able to use sets for abstraction of concepts to let us ignore details we don't care about.  For example, the field of set theory eventually defines the natural numbers as sets: they define $0 = \{\}$, $1 = \{0\}$, and $2 = \{0,1\}$.  You shouldn't have to care about that right now, but what if someone asks you whether $0$ belonged to the set $\{1, 2, 3\}$?  With non-recursive $\in$, you can immediately answer "no", but what if we had used the recursive belonging relation?  In that case the answer would be "yes", because $0 \in 1$ and $1 \in \{1, 2, 3\}$.
Using the current (non-transitive) belonging relation means when we use sets to define composite objects, we can use the resulting things as mathematical objects with their own properties instead of having to care about the details of how they were built out of sets.
