I wish close-voters would chime in as to what seems too broad; this strikes me similar to almost any optimization question: to definitively answer you'd have to consider lots of combinations, but that's always the case. No?

Should this build be viable, or just having as many dice in one fight as possible? For instance, throwing a fireball into themselves definitely adds one more die for the saving throw, et cetera

It should not be a burden on the party but it doesn’t need to be optimized to the max.

What do you mean by "for their attacks"? Damage dice only? Does being able to attack multiple times count?

@Szega It does, if each attack uses lots of dice :).

Sorry, but that does not answer either of those questions. How do you define many? Two attacks with 2d6 damage each are better or worse than one with 3d6? Only dice used for damage count?

@nitsua60 the criteria for the question is not well-defined, as you can infer from all the comments back and forth from the community the OP. This kind of question is too subjective, several builds could throw the same amount of dice and fits more a forum or reddit thread than a stack Q&A.

To make this question answerable, the judgement criteria need to be clear. For example, most dice thrown, regardless of size or reason, per-round, with the requirement that at least that many dice could be thrown each round for 5 consecutive rounds and then another 5 consecutive rounds an hour later. Alternatively, most dice thrown on a single attack, considering both the attack and damage rolls, with a build's score being the square root of the number of dice rolled to attack squared plus the number of dice rolled for damage squared, with die size being irrelevant

Also 'not a burden to the party' is similarly not sufficiently defined.

I'm not sure if this question has a definitive answer, particularly sine the criteria are not established. Level can make a huge difference. A fifth level Wizard throws a lot of dice for a fireball in one round, but might not last long enough to cast another, while a barbarian in rage might last long enough to toss quite a few d12's if he lasts 10 rounds due to his hp and rage damage resistance. If just number of dice are the only criterion, a character could simply close his eyes during combat and roll twice as many 2d0's on every attack due to disadvantage. It needs more constraints.

I find this question to be way too broad. It's like saying "what build of anything does the most damage?" It's always going to be situational esp. wrt number of opponents on the field.

@keithcurtis in the question, it already says it shouldn't be for a nova (once) but multiple times during the adventure day.

@KorvinStarmast No, I haven't because she was clear she wants to throw many dice in combat, and I also wrote that this is for attack. I clarified it now.

@thedarkwanderer The judgement criteria is the number of dice that are regularly rolled over an adventure day. I explicitely wrote it's not for a single attack but 'all the time', e.g. over the whole adventuring day.

@keithcurtis It does need more constraints, but rolling with self-imposed disadvantage is just the single most important part of any right answer to this question-- if you don't have advantage, self-inflict disadvantage and you roll twice as many dice. That part doesn't need clarification or nothing.

@thedarkwanderer updated the question to rule out self inflicted disadvantage.

You can self-inflict disadvantage without being a burden to the party. In any case, you need to figure out what you want before we can reopen this, I think.

@thedarkwanderer If you have a build that uses disadvantage to make the character stronger or better, please answer. Otherwise, I think it's clear that not weakening oneself is included in 'is not a burden to the party'. No? Anyway, clarified.

OK, that's all I've got, edited the answer a little bit for organization and hoping that it both addresses the combat issue and other chances to roll dice. Won't comment on the "build in a disadvantage to make the character stronger or better" since that style isn't as pronounced in 5e as it is some other game systems.

@KorvinStarmast Disadvantage is a very specific mechanic in 5e (roll 2 d20, take the worse result), so this was not meant in a general 'include a disadvantage' way.

I think this question would be improved if the composition of the rest of the party was added, since the "get a lot of rolls" can be influenced by who else is in the party when exploiting synergies ...

To all, the disadvantage mechanic was an example ad absurdum to show that the question needs serious constraints, not an actual suggestion.

@keithcurtis How is using Disadvantage to get more dice compatible with the outlined constraints? Intentionally being worse is definitely covered under not being a burden.