Is there a philosophy which argues that nothing exists?

I'm surprised this wasn't mentioned before, at least not explicitly (it was implicitly pointed out in one of the comments, as this view is considered to be nihilistic):

“Nothing exists; even if something exists, nothing can be known about it; and even if something can be known about it, knowledge about it can't be communicated to others.” - Gorgias

One must be careful though:

This argument has led some to label Gorgias as either an ontological skeptic or a nihilist (one who believes nothing exists, or that the world is incomprehensible, and that the concept of truth is fictitious). But it can also be interpreted as an assertion that it is logos and logos alone which is the proper object of our inquiries, since it is the only thing we can really know. On Nature is sometimes seen as a refutation of pre-Socratic essentialist philosophy (McComiskey 37). (source: IEP)

As many have said before me here, I think the question is self-defeating. You have the thought (i.e. it exists) to ask yourself whether there is a philosophy that asks whether nothing exist. The existence of the thought means that something exists and would famously give rise to Descartes' new foundation of all knowledge.

We join spokes together in a wheel,
but it is the center hole
that makes the wagon move.

We shape clay into a pot,
but it is the emptiness inside
that holds whatever we want.

We hammer wood for a house,
but it is the inner space
that makes it livable.

We work with being,
but non-being is what we use.

– Tao Te Ching, Stephen Mitchell Translation

I may have misunderstood it. I'm talking about philosophy where nothingness has a central role.

The statement "X exists" is meaningless without first defining "existence".

For nothing to "exist", that would mean that you and everything you know falls outside of "existence".

Any definition of the word "existence" where everything you know would not "exist" would be a rather absurd and meaningless definition, however.

I can't think of any philosophical movement that uses such an meaningless definition of "existence". However, there are philosophical movements who believe that everything that does exist is inherently absurd and void of meaning.

Examples of such movements would be Existentialists and Discordians in the West and the absurdist schools of Taoism & Zen in the East!

In the second century BCE Nagarjuna proved that nothing really exists and nothing ever really happens. This is the claim of Middle Way Buddhism and phrased in one way or another is the claim of the Perennial philosophy.

This is not nihilism but the claim that all division and distinction is unreal or reducible in metaphysics. To exist is to 'stand out', thus for something to exist there must be a background from which it stands out. This is two things, and a fundamental theory must reduce them.As Schrodinger points out, as well as what appears to exist there is the 'canvas on which they are painted'.

For the mystic the world reduces to the 'I Am' of pure awareness, which is rather like Kant's 'thing-in-itself'. This would transcend the distinction between existence and non-existence such that we cannot say it exclusively exists or does not. Hence the qualifier 'really' in 'nothing really exists'. This explains Heraclitus' curious statement 'We are and are-not'.

The logical impossibility of explaining why there is anything at all undermines the very idea that there is anything at all. Once we reify even one thing we cannot solve the metaphysical problems that arise, and this should tell us something.

That nothing exists is posited in medieval ontology

In this attempt mysticism arrives at a peculiar speculation, peculiar because it transforms the idea of essence in general, which is an ontological determination of a being, the essentia entis, into a being and makes the ontological ground of a being, its possibility, its essence, into what is properly actual.

This mystical theology appears to be saying, (amongst other things), that the essence of nothing does not require production to become actual. For example, whereas the concept (res essentia) of any other thing, such as a table, requires the making of the thing to become an actual thing (res existentia), 'nothing' does not require anything further to reify its concept. It uniquely qualifies as actual nothing by itself.