Taking no starting items

Long Swords and Boots are Starting items for a reason. You can start with these + a few pots and accelerate your build but champions are balanced around Doran items etc. to start, so youre sacrificing regen + damage for a pittance of gold and its almost never worth it.

Long sword + 3 pots OP

If you don't want Dorans why not go for long sword + pots or boots + pots or "crystal item" + pots or book + pots? All of those are much better than "no items"

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Is it a viable strat to take no starting items and keep your 500 gold for your first back, if you don't lose lane early on?

In the almost-impossible case that you don't lose lane immediately because of this, it's still not going to be worth it because of how much you're giving up in terms of early-game stats. It's not just that farming is harder, fighting becomes extremely difficult because of the stat advantage your opponents will have, and you'll be mostly dead weight throughout the lane. Without any regen (Pots, Doran's items), you'll easily get pushed out of lane and forced to take an early back, completely defeating the purpose of this.

Short answer: No, not in the slightest. If you really don't want to start with a Doran's item, AD characters can start Longsword + 3 Pots, AP can take Dark Seal + 3, bruisers/tanks can go Rejuv Beads.

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Is it a viable strat to take no starting items and keep your 500 gold for your first back, if you don't lose lane early on?

Even if you choose not to take the standard starting items, 500 gold will usually get you part of whatever you are planning for your first item. A partial bonus is still better than no bonus.