Could a planet, with life, exist without plants, but instead with fungi? How large could these fungi get? What would all of the conditions involved be?
Fungi, like animals, are heterotrophs, which means that they need to consume another organism in order to obtain their energy and nutrients.
Autotrophs, such as plants, are organisms that can utilize energy from a primary energy source and process nutrients in the environment in their raw form. They are the basis of a food chain and nothing can exist without them. If there are no plants then there needs to be some other type of autotrophic organism for the fungi to feed on.
My understanding is that we only know of two autotrophic mechanisms: photosynthesis which uses light and chemosynthesis which uses chemicals. So the pickings are slim unless your world has naturally occurring methane oceans and oxygen or some vast deposits of some other chemicals that can be reacted to release energy (which would probably count as chemosynthesis). The first examples of chemosynthesis we discovered earth react hydrogen with sulphur that comes out of geothermal vents.
I think most examples of chemosynthesis that we have found are closely tied to geothermal energy, but my understanding is that these organisms use the chemicals from the geothermal vents rather than the heat from the vent itself. Theoretically, I suppose an organism could evolve to directly use the heat differential from the geothermal energy and I guess these would be thermotrophs. I guess it is also theoretically possible that an organism could use the radiation produced from the decay of naturally occurring deposits of radioactive materials which would classify them as radiotrophs. But if they use the heat from the decay rather than the radiation itself then these would be thermotrophs.
EDIT: One thing that I think is theoretically plausible that the answer in DWKraus's link mentioned is electricity. I suppose if you had very large rocky deposits of the right material bridged by an ocean of the right composition it could form a giant battery with ions and electrons moving in the currents.
Reading the answer that DWKraus pointed to also made me think of an potential issue related to size which seems to be what you are interested in: Organisms start evolving small so if the energy gradient is spread over a wide area, the tiny originator organism cannot span enough of the gradient to obtain enough energy to survive unless the gradient is very intense. You would almost need an organism to evolve using an energy source effective over small gradients until it got larger and then evolve into a different energy source that was only effective over large gradients.
Could a planet, with life, exist without plants, but instead with fungi?
As long as you have some replacement for the autotrophic functions of plants, sure. Fungi can easily perform the structural functions of plants in building biomes.
How large could these fungi get?
As big as plants get. So, pretty much as large as you want.
What would all of the conditions involved be?
Since fungi don't perform photosynthesis, you would have to rely on bacteria / algae analogues for that, to provide the foundation to the food web. If you posit a scenario like early Earth, but in which vascular plants merely happen not to evolve, algae and bacteria will face the same kinds of competitive pressures to shade each other out for access to more light. And since they can't grow tall structures themselves, they will have to hitchhike on tall-growing fungi. Turns out, there are already extremely common examples of symbiotic relationships between fungi and bacteria and algae on earth: lichens. Earth lichens are small and slow growing, but without competition from vascular plants, they could easily fulfill all the same functions.
On Earth, plants rely on symbiosis with fungi anyway, with the plants providing the above-ground structure and feeding sugars to soil fungi, while the soil fungi extract mineral nutrients to provide to the plant. Your planet will simply shift the structural balance, with the fungal component forming the bulk of the structure and hosting photosynthetic symbiotes in its stems and leaves, rather than hosting nutrient-extraction symbiotes in its root systems.