This is kind of a spin-off of a question I did about a space telescope surveying the galaxy from above the galactic plane. So, it made me wonder, which object in the Milky Way has been spotted orbiting furthest above or below the galactic plane? I can't seem to find anything mentioning this. However, I have come across galactic/interstellar extinction, so I believe it's possible this question is just not knowable from our current location. On the other question, there is a link to a page that says our solar system is about 100 light years above/below the galactic plane. So, how far is the furthest object from the galactic plane spotted, to date?
Bochanski et al. (2014; "The Most Distant Stars in the Milky Way") reported the discovery of two stars -- ULAS J001535.72+015549.6 and ULAS J074417.48+253233.0 -- found to be M giants at distances of 274 ± 74 and 238 ± 64 kpc, respectively, "making them the first MW stars discovered beyond 200 kpc."
One might object that their distance estimates (based on using spectroscopy to infer their absolute magnitudes) are rather uncertain, and so another set of candidates comes from a recent paper (Medina et al. 2018) looking for distant RR Lyrae variable stars, which are one of the classic standard-candle distance estimators. Their Table 1 lists three RR Lyrae stars with distances $> 200$ kpc; the most distant is HiTS104738+020627, at a distance of $262.2 \pm 24.3$ kpc. Formally, this is not as far as ULAS J001535.72+015549.6, but the distance is better determined.
But you asked about the height above/below the Galactic plane, which is not the same as the distance. To determine the height, you need to know two things: the distance to the object, and its Galactic latitude (which can be determined from its equatorial coordinates). This is because the height is $h = d \sin b$, where $d$ is the distance and $b$ is the Galactic latitude.
From that, we can get the following estimates
|Object||d (kpc)||h (kpc)|
|ULAS J074417.48+253233.0 (M giant)||238||91|
|PSO J174.0675-10.8774 (globular cluster in James K's answer)||145||108|
|HiTS110510-022710 (RR Lyrae)||219||170|
|HiTS102014-042354 (RR Lyrae)||233||191|
|HiTS104738+020627 (RR Lyrae)||262||204|
|ULAS J001535.72+015549.6 (M giant)||274||$-$237|
So the winner is ULAS J001535.72+015549.6, at a height of 237 kpc (773,000 light years); the minus sign means it's "below" the plane rather than above it. If you want a more certain estimate, then HiTS104738+020627 at a height of 204 kpc (665,000 light years) is the best candidate.
Note that interstellar Galactic extinction is a problem for lines of sight in/near the Galactic plane, and especially in directions toward the Galactic Center. So it's not a major problem for objects located far above or below the plane, because when looking at one of those, you're only looking through a very small part of the Galactic disk near the Sun, where the extinction is mostly pretty small.
Globular clusters orbit in the Galactic halo, and so are good candidates for being far above the disc. The globular cluster PSO J174.0675-10.8774 is the furthest, 470,000 light years distant, well out in intergalactic space (aka Laevens-1, although Laevens recognises that his team was the second to discover it).
There are also dwarf spheroidal galaxies such as Leo-V that orbit at about the same distance.
Deciding exactly which is further is rather tricky, as exact distance measurements are not possible for objects such as these, that don't show any parallax. Moreover, both these objects are recent discoveries, and so it is likely that there are other, more distant, faint globular clusters and dwarf galaxies that are loosely bound to the Milky way but haven't been detected yet.