How to render a large, seamless platforming world without scene transitions?

Modern hardware is powerful and platformers are often pretty lightweight, so keeping the whole game world for a 2-3 hour game in one single scene might not be as infeasible as it might seem. So I would recommend you to just start building your game in one scene until you actually notice problems with low framerate or excessive RAM use... or if you have multiple level designers and need to enable them to work on different parts of the game world in parallel.

When you actually do have the problem, then there are a couple things you can do:

• Deactivate objects which are more than a couple screens away from the camera and activate them when the player comes into range (usually doesn't do much for RAM but can conserve some CPU and GPU instructions).
• Convert your game into sections which you load and unload dynamically when the player comes into range. Those sections can be represented by prefabs or by scenes you load with LoadSceneMode.Additive (as in the answer by Xamtouo)
• Use scene transitions, but hide them well. Place your scene transitions in relatively "boring" areas with no moving objects. Duplicate these sections exactly in both scenes. When the player moves from one scene to another, make sure their position and state get accurately transferred to the next scene. If you do it well, you can hide any indication that there was a scene switch.

Well, basic setup of desired configuration can be achieved quite easily in Unity.

SceneManager class has nice methods: LoadSceneAsync and UnloadSceneAsync. They are loading/unloading scenes in the background and provide convenient callbacks when scene loading is complete; you can initialize more than one scene like this:

 SceneManager.LoadSceneAsync("DesiredSceneName", LoadSceneMode.Additive);

This way already loaded scene stays untouched and the second scene is loaded in additive mode. If there are more than one scene initialized, all GameObjects from all scenes are displayed. You have options to Merge Scenes or to Move GameObject from one Scene to another. DontDestroyOnLoad is also nice for objects continuously traveling between scenes.

So the game world is split into different scenes, which are loaded in background and seamlessly linked together. How to arrange scenes loading/unloading is up to you:

• You can set up triggers near scene borders, which start loading adjacent scenes when entered.
• You can automatically load neighboring scenes when player crosses scene borders and unload all others so at any time you would have 3 scenes loaded: current, previous and next.

Another important note is that additive scenes might overlap each other, so you'll need to offset objects on each scene or write a script which places scene gameobjects in required location.

(For example, there is popular trick: move all UI to separate scene in additive mode to keep other scenes as clean as possible; UI must not be destroyed when scene is changed after all)