All classes get two saving throw proficiencies at level one and you can take Resilient to get another, but you can only take it once. Is there any other way to get another saving throw proficiency? My gut says no, because saving throws are pretty important.
All classes get two save proficiencies to start with, and can take the feat Resilient for a third. You can only take any given feat once, unless it explicitly allows taking it more than once (like Elemental Adept).
- Monk gains proficiency in all saves at L14 via Diamond Soul.
- Rogue gains proficiency in WIS saves at L15 via Slippery Mind, but makes no provision for choosing an alternate if you somehow already have it.
- Samurai Fighter (from Xanathar's Guide to Everything) grants proficiency in WIS saves at L7 via Elegant Courier and allows you to pick INT or CHA if you already have WIS.
- Gloom Stalker Ranger (from Xanathar's Guide to Everything) grants proficiency in WIS saves at L7 via Iron Mind and allows you to pick INT or CHA if you already have WIS.
- Transmutation Wizards can gain proficiency in CON saves at L6 via Transmuter's Stone.
If you're purely looking for big saves, there are other factors.
- Paladin adds CHA to all saves from L6 onward via Aura of Protection. It's not proficiency, but it can be better at low levels and stacks at all levels.
A character can get all six attribute save proficiencies at L14 via Monk. Monk is the only way to get proficiency in death saves (it says "all", which includes death).
Alternatively, a Gloom Stalker 7/Samurai 7 (also L14) with Resilient has five attribute save proficiencies. If you start as a Ranger, you can add Transmuter 6 to get all six. (You don't want to start as Fighter, because they get Constitution naturally, and you want to get it from the stone.)
An L6 Transmuter or an L15 Rogue with Resilient comes in last in the big-saves contest with four and no built-in way to get the Paladin's bonus for the Rogue. I'd put the Transmuter a little bit behind, because it gives up on all the other things the Stone can do if you're using it just for the saves.
Anybody else can have three, via Resilient.
If you had a generous Transmuter friend, you could borrow his stone to add Constitution to any character, but the Transmuter loses out on it himself.
Might not be the answer you want, but Monks gain proficiency in all saving throws at level 14 from their diamond body trait. Rogues gain proficiency in Wisdom saving throws at level 15 from their slippery mind trait. This list is not conclusive but I don't have my books with me.
Some classes get an additional saving throw. For example, the UA Ranger Deep Stalker conclave (included in Xanathar's Guide to Everything as the Gloom Stalker conclave) at seventh level earns proficiency in Wisdom saving throws. When you include the Resilient feat, it is possible to get four proficient saving throws.
Also, it should be noted that the new UA Artificer revisited class basically gives a 20th-level Artificer the equivalent of proficiency in every save. The Soul of Artifice feature allows them to be attuned to up to 6 items at once. For each item they are attuned to, they gain +1 to all saves = +6 to all saves potentially. Since this gets added to all saves, on top of any proficiency bonuses, it is even better than giving them proficiency in every saving throw. Obviously, the really big downside is it's only at level 20 that they achieve this, and most players never reach level 20.
There's nothing to say that you can't take the Resilient feat more than a single time. Let's use Clerics, for example. They start with Wis and Cha as their saves. If you're building a front lines war cleric, you may be working str and con as main stats as well. At level 1 have each of those stats being an odd number 11, 13, etc. When you gain the chance at an ability score upgrade or feat at 4th level, choose Resilient with that selected ability score. Using str as an example. At level 1 have 13 str, at level 4 gain +1 to str and proficiency in saves for it. You don't take as hard of a hit by not gaining the two ability score increases at that level, by burning a feat instead that is still quite useful. Then do it again for Con next time you get an ability score increase and you now have 4 saving throws you are proficient with.