Can I have any number of Relentless Rats in my sideboard, regardless of whether or not I'm playing any in my main deck? Does the definition of deck include the sideboard?

Two separate definitions, so no. Relentless Rats does not have text in direct contradiction of the Comp Rules regarding sideboard quantity.

The more interesting question, though, is: If your deck has four or more Relentless Rats in it, are you allowed to put any Relentless Rats in your sideboard at all?

I would argue that Rats has text contradicting rule 100.2a, and since 100.4a just invokes 100.2a, Rats' text applies just as well there. there's no separate "Sideboard can only have up to four of a card, minus the number in the main deck" rule, it just says "hey, that limit set out elsewhere applies to both things together."

I just realized I may have misread the initial question. For some reason I thought he was asking if he could have a sideboard of 16+ cards if they were Relentless Rats.

oh, that's possible. in that case, you're right, 15 is still the limit.

I think the usual question that follows is "Why is it assumed that Relentless Rats / Shadowborn Apostle supersedes 100.2a and the second sentence of 100.4a but not the first sentence of 100.4a"? The card says you can have any number. Why can't I have 129 of them in my sideboard? Because the rules say I can't? But the rules also say I can't have more than 4 in my sideboard, and the fact that this tells me I can have any number allows me to break _that_ limit, why not the 15 limit too?"

I think it'd be silly to allow the 15-sideboard limit to be breached. But I don't think they've covered off the logical cases here to be able to say that's implict in existing rules when this is, by definition, a rules-overriding clause.

The rules don't need to be immune to misinterpretation. That's an unreasonably high standard, both since people can be very good at misinterpreting things when they put their mind to it and since trying to make it immune to that would also make it much harder to read and understand. The purpose of the rulebook is to allow players and judges to know how the game works and to ensure that judges make consistent rulings. So the possibility that someone could argue that "technically, the rules could be interpreted to mean [some absurd outcome]" isn't relevant; that sort of rules lawyering doesn't work.

Also, when a card conflicts with the rules, it overrides only the specific rules in question, not any tangentially related ones. The ability "A deck can have any number of cards named Foo" doesn't mean "There can be absolutely nothing restricting you from putting as many cards named Foo into your deck as you want", but "The usual rule limiting how many cards of a given name you can have in your deck (100.2a) doesn't apply to Foo".

I would venture that the 15 card limit of a sideboard is not superseded by the text of Relentless rats/etc. I base this opinion on the fact that the ruling for Commander (another max card limit example) allows up to 99 Relentless rats and a Commander (though, as Sheldon said, you might want some way to cast them).

A normal deck has no maximum card limit, hence no limit on the number of rats since you can have more than four. But since Commander does have a max card limit and Rats observe that limit then it seems reasonable to conclude that rats would observe the max limit on a sideboard.

My question is not to go over 15 cards but for more than 4 of those 15 to be relentless rats

Razor already answered this question with a "yes, you can put up to 15 Rats in your sideboard." Albeit using different words. And in response to my misunderstanding of the question where I quoted the rules and said the sideboard is still limited to 15 cards.