In fact, for a long time the expectation was that the human Y chromosome would lose all function in ~10 million years - now the expectation is that it will take longer.
From Wikipedia, the Y chromosome (I have reordered the paragraphs, emphasis mine):
In humans, the Y chromosome spans about 58 million base pairs (the building blocks of DNA) and represents approximately 2% of the total DNA in a male cell.
The human Y chromosome contains 86 genes, which code for only 23 distinct proteins. Traits that are inherited via the Y chromosome are called holandric traits.
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By one estimate,
the human Y chromosome has lost 1,393 of its 1,438 original genes over the course of its existence, and linear extrapolation of this 1,393 gene loss over 300 million years gives a rate of genetic loss of 4.6 genes per million years.
Continued loss of genes at the 4.6 genes per million year rate would result in a Y chromosome with no functional genes --- that is the Y chromosome would lose complete function --- within the next 10 million years. Comparative genomic analysis, however, reveals that many mammalian species are experiencing a similar loss of function in their heterozygous sex chromosome. Degeneration may simply be the fate of all nonrecombining sex chromosomes due to three common evolutionary forces: high mutation rate, inefficient selection and genetic drift. Furthermore, comparisons of the human and chimpanzee Y chromosomes (first published in 2005) show that the human Y chromosome has not lost any genes since the divergence of humans and chimpanzees between 6–7 million years ago, and a scientific report in 2012 stated that only one gene had been lost since humans diverged from the rhesus macaque 25 million years ago.
These facts provide direct evidence that the linear extrapolation model is flawed and suggest that the current human Y chromosome is either no longer shrinking or is shrinking at a much slower rate than 4.6 genes per million years estimated by the linear extrapolation model.
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Future evolution
In the terminal stages of the degeneration of the Y chromosome, other chromosomes increasingly take over genes and functions formerly associated with it.
Finally, the Y chromosome disappears entirely, and a new sex-determining system arises. Several species of rodent in the sister families Muridae and Cricetidae have reached these stages, in the following ways:
The Transcaucasian mole vole, Ellobius lutescens, the Zaisan mole vole, Ellobius tancrei, and the Japanese spinous country rats Tokudaia osimensis and Tokudaia muenninki, have lost the Y chromosome and SRY entirely. Tokudaia spp. have relocated some other genes ancestrally present on the Y chromosome to the X chromosome. Both genders of Tokudaia spp. and Ellobius lutescens have an XO genotype, whereas all Ellobius tancrei possess an XX genotype. The new sex-determining system for these rodents remains unclear.
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Outside of the rodent family, the black muntjac, Muntiacus crinifrons, evolved new X and Y chromosomes through fusions of the ancestral sex chromosomes and autosomes.
Primate Y chromosomes, including in humans, have degenerated to an extent that primates could also evolve new sex determination systems, by some estimates in about 14 million years in humans.
Contrast this with the X chromosome which, in humans, spans more than
153 million base pairs (the building material of DNA) and
represents about 2000 out of 20,000 - 25,000 genes.
Human sex may not become extinct...but it may become very different!
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