"Some people will be very disappointed if there is not an ultimate
theory that can be formulated as a finite number of principles. I used
to belong to that camp, but I have changed my mind. I'm now glad that
our search for understanding will never come to an end" -Stephen
Hawking in his lecture Godel And The End Of Physics.
The Penrose-Lucas argument goes
"while a formal proof system cannot prove its own consistency,
Gödel-unprovable results are provable by human mathematicians. He
takes this disparity to mean that human mathematicians are not
describable as formal proof systems, and are therefore running a
There certainly are people who disagree with this, but I think the ideas of a Nobel Prize winner with many substantial contributions to physics, should at least not be dismissed out of hand (though I fully expect other answers or comments to do so).
A unified field theory, or quantum-gravity, no problem. But a Theory Of Everything is intrinsically a goal arising from hubris (like the Hilbert programme, that Godel ended). It's not even clear precisely what it means, how it could ever be discrete or finished, it would have to include every aspect of knowledge and experience, with some degree of predictions.
But can we fully predict conscious beings, without making models that are, conscious? When a theory has conscious elements, it's a mind, or contains them.
I see Hofstadter's strange loops and tangled hierarchies, as the way out of Godel Incompleteness, by sidestepping the foundations for knowledge of the Munchausen trilemma: axioms cannot justify axioms, circular reasoning cannot justify circular reasoning (eg problem of induction), infinite regress is possible but must exceed a finite system of knowing, you could never verify it. So we cannot prove anything in a foundational way - only within a system. Strange loops point to how consciousness can arise within a system, as strange loop feedback, and through a process of tangling hierarchies build a process of coherentism, in which strange loops develop emergent capacities to condition the system rather than be conditioned by it. This has been identified as a repeated motif in philosophy, eg Wittgenstein's ladder. In this view proof systems are not out there, but within us, specifically within the collaborative intelligence of language. And with new words, new ideas, that changes, and new truths are possible.
A mind can begin where it is, and 'on the fly' generate a new system, in a way formal systems cannot. This is what avoids the halting problem and creates Godel incompleteness: a mesh of interacting priorities that can seek to highlight and solve problems with reference to an entire tangled hierarchy, inc adding layers. The truth unprovable in the previous system, is recognised as truth by a mind because it has changed the system (typically adding axioms).
So, a complete system is possible, but would involve creating a 'theory' that could continue to elaborate itself, ie a mind capable of thought. I'd link this to the emergent mutually-arising collaborative network of Indra's Net, rather than any prexisting monistic mind with foreknowledge. The universe coming to understand it's own self, not as a conditioned organism, but as a choice how to be.