What are the top 3 champions with the most dps
The title.
The title.

Do you mean ap champs or ad champs?
Well casio is up there

Some ADC
In general situations,
assuming they're performing their dps potential without interruption.
In extreme situations,
because they had infinite farms and one hit would deal oo damage.
If u meant burst instead of dps,
because of their stat scaling?

My main man 
This annoying fucker 
This lonely abandoned avenger.. 
Illaoi hands down tho
The king of dps is and will be karthus (with 1000+ap) Anyone who disputes that , please provide a champion contest with 6 item build and I will do the calculations to show you the dps differential over 20 Seconds.
All marksmen, except reloaders like Jhin and Graves.
Heads up guys, DPS stands for Damage Per Second. This means champions who deal damage, as opposed to healers, tanks, or disruptions. Burst damage is DPS. You're dealing massive damage over a short period of time. Time is still calculated in seconds, especially if a champion can take 1 or 2 to remove your HP bar. These specific champions you list such as Brand, Cassiopeia are DOT mages, and deal specifically in debuffs that deal damage, along with their burst. Karthus is similar, except his DOT is an AOE field around him. So get your terms right. DPS - Damage Per Second, literally every damage dealing class is DPS. DOT - Damage over Time, specifically poisons, burns, etc. A slow form of DPS. Burst - Burst damage, high damage dealt over a short period. The fastest form of DPS.
Probably a full AP
with a crap load of AP
For the 7 seconds his ult is active
with a fully stacked
and some % health dmg items. Possibly with a 
Karthus with flat magic penetration runes red masteries and 3 deathcaps sorcerer boots laundry's torment archangel staff. Run E and just farm minions.
Nobody else comes even close
in what part of the game? early? mid? late? area or single target? and how much does practicality matter? and what time window are we looking at? 5 seconds? 10? 20? 2?