Can an accelerated timelike trajectory reach the boundary of AdS space in finite time? I understand why in anti-de Sitter (AdS) spacetime, null geodesics can reach spatial infinity in finite coordinate time, while timelike geodesics cannot reach spatial infinity at all, not even in infinite coordinate time (in the sense that any timelike geodesic's radius is bounded).  But what about accelerated (i.e. non-geodesic) timelike trajectories?  Can they reach spatial infinity in finite coordinate time?  Finite proper time?  In not, can they come arbitrarily close, unlike timelike geodesics?
Intuitively, it seems to me that the case of accelerated timelike trajectories is intermediate between the case of timelike geodesics and the case of null geodesics, because an accelerated timelike trajectory can "escape from the origin better than" any timelike geodesic by accelerating against the AdS gravitational attraction, but it can't "escape from the origin as well as" a null geodesic because it can never hit the speed of light.  My intuition is that an accelerated timelike trajectory could reach spatial infinity, but only in both infinite proper time and infinite coordinate time.
But I'm not confident that I'm correct, because I don't have good intuition for the boundary behavior of AdS space.  The fact that all timelike geodesics are bounded away from spatial infinity suggests to me that it's "harder" to reach spatial infinity in AdS space than in Minkowski space.  But on the other hand, the fact that null geodesics can reach spatial infinity in finite coordinate time suggests that it's "easier" to reach spatial infinity in AdS space than in Minkowski space.  Clearly these intuitions conflict with each other.
 A: The issue of coordinate time vs. proper time/affine parameter is a bit of a red herring, since the former is a purely gauge (i.e. coordinate-dependent) quantity, while the latter is physical but uninteresting in the context of the asymptotic boundary.  For instance, you say that null geodesics reach infinity in finite coordinate time, but for a given null geodesic I could of course choose a time coordinate that diverges when the null geodesic reaches infinity, making that geodesic reach the boundary in infinite coordinate time.  Likewise, by definition any curve (of any signature) that reaches the asymptotic boundary does so in infinite proper time/distance/affine parameter.
Instead, the covariant observation is that null geodesics always reach the asymptotic boundary of AdS, while timelike ones never do.  Your question can then be stated covariantly as: do there exist timelike curves that reach the asymptotic boundary of AdS?  The answer is clear from the conformal diagram of AdS:

(ignore the two dots labeled $i^+$ and $i^-$; I just grabbed this picture off the web).  The asymptotic (or conformal) boundary of AdS is the vertical line marked $r = \infty$ (the line $r = 0$ is just one possible choice of origin), and the line moving at 45$^\circ$ is a light ray (shown here reaching the boundary and then bouncing back in).  With this picture, it's clear that we can draw a timelike trajectory (that is, a line whose angle with the vertical is always less than 45$^\circ$) to and from the conformal boundary, so indeed, there do exist timelike curves that do reach asymptotic infinity.
Edit: As Peter pointed out in a comment, a statement I made regarding proper time in my previous answer was slightly wrong.  I believe the correct statement is that any timelike curve that reaches the asymptotic boundary must either do so in infinite proper time (e.g. a uniformly accelerated observer can eventually reach the boundary, but takes an infinite proper time to get there) or have divergent geodesic acceleration there (e.g. a timelike curve can get "close enough" to being null that it reaches the boundary in finite proper time, but it takes an infinitely large acceleration to do so).  The physical interpretation, of course, is that no finitely-accelerated observer can reach the asymptotic boundary of AdS in finite proper time.
