How large could a solid planet be (in theory) without becoming a star or black hole? Too much in the way of light elements would lead to thermonuclear ignition, but too many heavy elements would eventually cause gravitational collapse.
Similar questions have been asked before, but in this case assume that the planet's composition can be adjusted to any natural material substances in any proportions.
The critical point here is not the mass. It is the mass density and the resulting gravitational force on the atoms your planet is made of. See this post about the requirements for a black hole The same basic idea holds for turning a planet into a star.
In detail: If you make up your planet, the more matter (and thus mass) you add, the stronger the overall gravity of your planet becomes.
The stronger the gravity becomes, the more your matter is compressed, increasing the mass density of your planet (and also pressure and temperature, eventually turning solids to liquid to gas to plasma...) increasing the kinetic force of your atoms.
If the gravitational and kinetic force of the protons exceeds the electrostatic forces which keeps your atoms pushed apart, you start a nuclear fusion reaction and your planet becomes a star. Wikipedia has all the details
Conclusion: As long as you do not exceed a gravitational density threshold your planet can be any size you like. The exact numbers will require a non-trivial amount of calculation for any given case.
However, if you are writing fiction and not a documentary, consider an artificial planet-like structure that is mostly hollow for extremely large sizes. This way you will prevent to much matter from concentrating on one point. The limit there is your stellar neighbourhood, with nearby stars starting to mess with the integrity of your construct through gravity.
The largest known planet so far seems to be about 1.7 times the size of Jupiter (Source), which is utterly large.
Of course if you need a solid or liquid surface, or human beings being able to live on it without aid, your planed has to be much smaller.
An alternative to a carefully engineered supergiant bubble or honeycomb structure is to just throw matter together in the right order to keep it from sustaining fusion in the core.
With that in mind, mass matters more than volume. Thus my answer is: about 1.4 solar masses. Just shy of the threshold of pressure to turn atoms into degenerate matter, and thus becomes a neutron star.
Since iron takes more energy to fuse than it releases, a body with a core made entirely out of iron can't undergo fusion. Start making your planet out of heavy elements until you have a huge ball that's mostly iron. Fission might happen if you throw very heavy elements into your planet in huge chunks, so try to mix things up first. Or stand well back.
After that, just start throwing in whatever is available. Hydrogen is cheap and plentiful in the universe, and since you're going for a huge planet, it's going to be a gas supergiant anyways, rather than a rocky planet; just with an iron core rather than a metallic hydrogen core like our solar systems' gas giants have.
There are many massive bodies of the universe that don't do thermonuclear burning. The problem may be that they are all classified as stars.
A naturally formed planet contains hydrogen (and deuterium) and can not avoid burning it at least a little if its mass exceeds 12 Jupiter masses. If there is no deuterium, though (naturally, it burns out) this object will look like planet and not a star - so we can technically call it a planet. If the mass of the object goes higher - to 65-80 Jupiter masses, it starts burning Lithium, and, ultimately, Hydrogen-1 - which means this object a "real" star.
But what if the object does not have any "combustible" elements? Naturally, this occurs when larger stars burn out - their core becomes iron, which can not fuse any more to produce energy. These objects are called "white dwarfs" (if their mass still under Chandrasekhar limit, which is about 1.4 Solar masses). They are still hot, though, and it takes trillions and quadrillions of years for them to cool down and become "Black dwarfs". Can we call "black dwarf" a planet?
Still, the mass can go higher. An object heavier than 1.4 Solar masses, but lighter than about 3 Solar masses (Tolman–Oppenheimer–Volkoff limit) will become a Neutron star. Natural neutron stars are even hotter than white dwarfs, and longer time would be needed to cool them down. Can we call a cool Neutron star a planet?
Unfortunately, it does not look like we can go any higher than 3 solar masses. This object would collapse into a black hole, which likely would not fit into any definition of a planet.
