In this article we explore the idea that consciousness is a language-complete phenomenon, that is, one which is as difficult to formalise as the foundations of language itself. We posit that the reason consciousness resists scientific description is because the language of science is too weak; its power to render phenomena objective is exhausted by the sophistication of the brain’s architecture. However, this does not mean that there is nothing to say about consciousness. We propose that the phenomenon can be expressed in terms of data compression, a well-defined concept from theoretical computer science which acknowledges and formalises the limits of objective representation. Data compression focuses on the intersection between the uncomputable and the finite. It has a number of fundamental theoretical applications, giving rise, for example, to a universal definition of intelligence (Hutter, 2004), a universal theory of prior probability, as well as a universal theory of inductive inference (Solomonoff, 1964). Here we explore the merits of considering consciousness in such terms, showing how the data compression approach can provide new perspectives on intelligent behaviour, the combination problem, and the hard problem of subjective experience. In particular, we use the tools of algorithmic information theory to prove that integrated experience cannot be achieved by a computable process.