Some people talk about the possibility of a planet 9. Could there be a planet 10 or 11? How many undiscovered planets could there be in our solar system?
Let's take a look at the 2006 IAU planet definition
A celestial body that (a) is in orbit around the Sun, (b) has sufficient mass for its self-gravity to overcome rigid body forces so that it assumes a hydrostatic equilibrium (nearly round) shape, and (c) has cleared the neighbourhood around its orbit.
a) This does not make much of a limit. While we can rule out any extra planets in the relative proximity to the Sun (out to the Kuiper belt or so), there's a vast amount of space out to about a light year were objects would still be orbiting the Sun, but be very difficult for us to observe.
b) Objects large enough to gravitationally round themselves are discovered with regularity. While it's hard to observe the exact shape of objects farther away than Pluto, we have many solar system examples showing a radius of a few hundred kilometres is sufficient to achieve hydrostatic equilibrium.
c) This is what gives new planets the most trouble. Far from the Sun, the region an object needs to gravitationally dominate far too great, and even large objects will have to share space with millions of rocks and ice fragments.
So there being more planets is unlikely due to candidates being far enough away to not having been discovered yet also being so far out that they have difficulty clearing their orbit.
That said, the planetary definition is arbitrary, and if we discover planet-sized objects on the very rim of the solar system (something we can not rule out due to insufficient observation capabilities), the definition will again come into question. In that case we could gain quite a large number of new planets.
Our current understanding of planetary formation is that planets form from a protoplanetary disk that thins out the further we move away from the Sun. We can see this by observing that Neptune and Uranus are much less massive than Jupiter and Saturn, and there are no gas giant planets farther out (as far as we know). So I don't think there's much to find anymore, but who knows.
There are two possibilities how we could find new planets there though:
- A gas giant was almost ejected by gravity interactions with other gas giants in the early solar system. This is called planetary migration and is kind of a hot topic because we know of so many exoplanets that are found too close to their star than we think it is possible if they formed there (Hot Jupiters).
There has been some discussion about a possible ninth planet that would be of comparable mass to Jupiter and Saturn and that could explain some comet orbits. If this planet 9 exists, it would probably have formed much further in and then have been ejected to its current orbit by such interactions.
- Small spherical bodies the size of Pluto or less are quite abundant there. The reason the official definition of a planet includes 'a body that has cleared its orbit' is that otherwise we would already have dozens of planets because there are a lot of spherical bodies orbiting the Sun, for example Ceres in the asteroid belt. If, for some reason, the IAU relaxes this rule a bit, this would give many bodies outside the Kuiper belt the chance to become a planet.
Maybe some day we will find Earth-sized bodies there that are too big to be called dwarf planets, and the IAU changes the definition to include them as planets (which I doubt tbh), or make a new subcategory of planets for them.