Could a star closely orbit a black hole long enough for the star to have lost 0.5B+ years to time dilation?
Yes, but not very likely.

The closest orbit that does not require constant expenditure of energy to maintain it is the prograde equatorial ISCO. For a Kerr black hole the time dilation factor on this orbit is $$\frac{dt}{d\tau}\approx \frac{2^{4/3}}{\sqrt{3}(1-a/M)^{1/3}},$$ which at the astrophysically likely Thorne limit $a = 0.998M$ gives a dilation of 11.5470. For a Schwarzschild black hole ($a=0$) it is merely 1.22. (Hamilton 2018, p. 654)

So a 13 billion year black hole could have a star orbiting that had just experienced 13/11.5470=1.12583 billion years!

Now, the gravitational wave emission from orbiting like this would no doubt sink the star. After all, stellar remnants starting at least part of an AU apart are merging today, and the emission scales up fast as one gets closer. Ergosphere processes might happen, but they also reduce $a$ and hence the time dilation.