Sooooo, who are the most/least item dependent in top, mid, bot, jungle? :3
Main reason mages aren't the most item-dependent champions in the game is that pretty much every mage deals ~1000 base damage with their QWER combo @18.
To deal double damage you need ~450 AP: which is a fairly difficult within a typical 33 minute 11,000 gold-per-player game, making heavy itemization on mages capable of not much beyond doubling their offensive power.
That said: there's a major difference between OHKOing an opponent with a QWER vs. dropping them to 10%, which is likely why the scaling is so slow with mage itemization. Getting that extra 10% offers a considerable power-spike vs. not-having it.
In the pro-scene mid-lanners also tend to get the most gold and most farm of anyone though, so I wouln't just let someone talk you out of it.
For heavy item dependency:
The more AA reliant a champion is and the less burst they get in their QWER, the more item dependent they tend to be.
Unlike the 1000 QWER damage @18, the average level 18 AD is ~100, and ~110 for most AA heavy champions, with a 1.0 second AS.
With just 110 AD such champions double their AA DPS to 220, which isn't that far off from 1 item.
WIth 150% AS (due to how AS scales), AA heavy champions double their AA DPS again, from 220 to 440.
With 66.67% Crit along with IE's crit damage amp, AA champions double their AA DPS again, from 440 to 880.
With 50%+50 armor penetration, AA champions effectively double their AA DPS to the equivalent of 1760 (assuming 100 armor / 50% damage reduction on opponents).
Marksman are generally the most item-dependent class due to the exponential scaling avalible from itemization, though the consistancy provided by good base damage and a solid QWER usually makes the caster-heavy, less heavily scaling marksman better and more reliable within any given meta, favoring half-mage marksman like Lucian, Ezreal, Corkie, and Graves more than committed AA marksman like Vayne, Tristana, Twitch, and Jinx.
As for no item dependency:
Massive utility is usually what causes that.
Pretty much every damaging skill in LoL has itemization scalings tied to it, where a 2 second stun is generally a 2 second stun regardless of what items are or are not purchased.
As such, kits with a ton of utility and supportive qualities tend to be the ones reliant on the least amount of items to function properly in late-game, due to CDR often being the only stat that makes any kind of difference in the midst of combat.
Another factor with no item dependency is that a champion is simply all about early-game, a phase when hardly anyone has any items and as such scaling means very little.
Such champions aren't item reliant more due to there time to shine being before items can really be had over there interactions with scaling, which is still technically being really good without items.
Sooooo, who are the most/least item dependent in top, mid, bot, jungle? :3
Most item dependent in these lanes would be (or at least these are in the top 5 for the lane):
Top
Jungle
Mid
Bot
Support
Least item dependent (or bottom 5 for the lane):
Top
Jungle

Mid
Bot

Support