This compound is called [7]circulene and is one of the examples of non-planar aromatic systems (1). From the first overview of this compound, it is observed that there are 7 benzene rings so one would conclude that it is an aromatic compound:
There are $7 \times 4 = 28$ $\pi$-electrons, so this compound satisfies the criteria of anti-aromacity. Then how come this compound is aromatic?
(1) Smith, M. March’s advanced organic chemistry: reactions, mechanisms, and structure, 7th ed.; Wiley: Hoboken, New Jersey, 2013, p. 48, compound 40.
Answer
TL;DR: We don't count $\pi$ electrons here, but the $\pi$ conjugated circuits (1). [7]-сirculene has all conjugated circles of size $4n + 2$, and none of $2n$, what makes it an aromatic compound.
Strictly speaking, the aromaticity $4n + 2$ and antiaromaticity $4n$ rules in their original interpretation are very limited and cannot be universally applied:
According to the original Hückel $4n + 2$ rule, if a monocyclic, planar molecule has $4n + 2$ $\pi$ electrons, it is considered aromatic.
But the count of $\pi$ electrons is not what matters for polycyclic compounds, therefore a generalized $4n + 2$ rule exists: if a conjugated molecule has only $4n + 2$ $\pi$ conjugated circuits, it is considered aromatic.
Non-benzenoid compounds can have both $4n + 2$ as well as $4n$ number of conjugated circuits counted by taking into account all Kekule structures. Among non-benzenoid non-alternant polycyclic hydrocarbons [5]-сirculene and [7]-сirculene both have only $4n + 2$ ($6, 10, 14, ...$) numbers of circuits.
(1) Gutman, I.; Cyvin, S. J. Journal of Molecular Structure: THEOCHEM 1989, 184 (1), 159–163.
Also, check out this question regarding pyrene.
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