The thickness of the planetary disc is dominated by Neptune, due to its large orbital radius.

We can calculate a planet's maximum distance from the ecliptic $h$ from the inclination angle of its orbit $\theta$ and its aphelion distance $r$. We get a right triangle, with $r$ as the hypotenuse, so
$$h = r\sin\theta$$

The table below was calculated using data from the NASA [Planetary Fact Sheet](https://nssdc.gsfc.nasa.gov/planetary/factsheet/). Angles are in degrees, distances are in millions of kilometres.

### Planet distance from the ecliptic plane.

|Name | Inclination | Aphelion | Distance|
|-|-|-|-|
|Mercury | 7.0 | 69.8 | 8.506|
|Venus | 3.4 | 108.9 | 6.458|
|Earth | 0.0 | 152.1 | 0.000|
|Mars | 1.9 | 249.2 | 8.262|
|Jupiter | 1.3 | 816.6 | 18.526|
|Saturn | 2.5 | 1514.5 | 66.062|
|Uranus | 0.8 | 3003.6 | 41.937|
|Neptune | 1.8 | 4545.7 | 142.784|

So the total thickness of the disc is $2×142.784 = 285.568$ million kilometres, which is almost $1.91$ au.

---

Here's the Python code I used to create that table:
~~~
from math import sin, radians

names = (
 'Mercury', 'Venus', 'Earth',
 'Mars', 'Jupiter', 'Saturn',
 'Uranus', 'Neptune',
)

# Orbit data from https://nssdc.gsfc.nasa.gov/planetary/factsheet/
# Inclination to ecliptic plane
inc = [7.0, 3.4, 0.0, 1.9, 1.3, 2.5, 0.8, 1.8]
# Aphelion in millions of kilometres
aph = [69.8, 108.9, 152.1, 249.2, 816.6, 1514.5, 3003.6, 4545.7]

print("|Name | Inclination | Aphelion | Distance|")
print("|-|-|-|-|")
for n, th, r in zip(names, inc, aph):
    # Perpendicular distance to eciptic
    h = r * sin(radians(th))
    print(f"|{n} | {th} | {r} | {h:.3f}|")
~~~

Here's a [live version of the script](https://sagecell.sagemath.org/?z=eJxVUsFu2zAMvesriPQQp_BkJ3bSJMAOA7bDBqwdMGyXoQdOlithtixQ9ICu7r9XUoMNkwCJfBQfnkj2NI0wIhuwo5-IIVhXAmFn0QUhHI46wFsoBKw_a1IzPa5LWH_Xbg7J-IDEZl2mKFJGPs3esqZkfkWeyeXoN8JLxq32PDsd0Y0QV3BHPy1Dh4zQJymG2YdzVbkQOiUfQq-kw4DyYfpd-QGdZqTHqkfFwWjNVaT46NRgHbKdHPAEOnqerYL8XFinov4fN7IuoZFtCXWytvKUjqaEndwn7Jjc432ke-eNHhKXdTDaIZkBph5-2WEaNZMOAr1JnIdTTquPmWy_k9tI157kroTj9iAPCdy2ib-p6yb57b7dy5t7ITxZx8VquY31heW_Lyz_FCzw3gZGp_Sy2vzNeXPZEeongtguNrFlSe8f64vcsjJ6qoQodHMWENcVfNHkteusmgck6C7ErxXLBcvv0scIrtMYFJcpKNhsNjn4qqBfLU_uOYp7YpMvyqc5y6Z_jqpeAIVXtGI=&lang=python) running on the SageMathCell server.

---

As John Holtz mentions in the comments, the true $h$ value for a planet may be smaller than the value shown in my table. The table's $h$ value only occurs if the planet's [argument of periapsis](https://en.wikipedia.org/wiki/Argument_of_periapsisis) is ±90°. Fortunately, Neptune's argument of periapsis is currently ~272°, so my $h$ value should be fairly close to the true value.

---

James K has supplied a list of orbit inclinations to the Solar System's [invariable plane](https://en.wikipedia.org/wiki/Invariable_plane). Here's the table using those values.

### Planet distance from the Solar System invariable plane.

|Name | Inclination | Aphelion | Distance|
|-|-|-|-|
|Mercury | 6.34 | 69.8 | 7.708|
|Venus | 2.19 | 108.9 | 4.161|
|Earth | 1.57 | 152.1 | 4.167|
|Mars | 1.67 | 249.2 | 7.262|
|Jupiter | 0.32 | 816.6 | 4.561|
|Saturn | 0.93 | 1514.5 | 24.582|
|Uranus | 1.02 | 3003.6 | 53.468|
|Neptune | 0.72 | 4545.7 | 57.121|

That reduces Neptune's $h$ considerably! Uranus may even be the current "winner", depending on their arguments of periapsis with respect to the invariable plane.