There aren't a lot of items that have components that you can't buy at least part of, so I doubt it. I guess you could try rushing an Infinity Edge by saving for a B.F. Sword.
It would need to be someone with survivability early game because you couldn't rely on hp regen from items or potions. Or if you got heal or clarity as summoner spells you could last longer.
Part of the strategy would be that other people might think you are noob, causing them to get greedy and perhaps tower dive.
Part of the strategy would be that other people might think you are noob, causing them to get greedy and perhaps tower dive.

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Would there ever be a useful situation where you start a game without any items in order to work towards a more expensive one?