If you try hard enough, there are enough sources of saving throw bonuses that you can actually have a character incapable of failing a saving throw (without penalties being applied). Here's a list of bonuses I've been able to find that either apply to all saves, or to death saving throws specifically.
Remember that bonuses of the same type don't stack, so you can't grab all of the items listed below and hope for +13 to your saves. However, many of the sources listed are untyped, which will stack together just fine.
- Crusading Zealot: +1 bonus to saving throws until you fail a saving throw (resets after a rest)
- Scorned Noble: +2 bonus to saving throws while no allies are within 5 squares of you
- Chevalier level 10: +1 power bonus to saving throws
- Hordelands Nomad level 10: +1 power bonus to saving throws
- Masked Lord level 10: +1 item bonus to saving throws while wearing your lord's armor
- Tuathan level 1 (Continue the Story): +1 bonus to death saving throws
- Born of Shadow: +1 bonus to saving throws while in dim light or darkness
- Courageous Mind (requires Psionic Human): +1 bonus to saving throws while bloodied (a dying character is by definition also bloodied)
- Death Scorned (requires level 11 Revenant with 13+ Constitution): +1 bonus to death saving throws, and you don't die due to failing death saving throws until you've failed (1+Con mod, max 5) times, rather than the normal 3.
- Die Hard (requires Human): If you have failed 2 death saving throws, get a +5 feat bonus to death saving throws and you can spend a healing surge at a result of 15, rather than the normal 20.
- Disciple of Death (requires 13 Wisdom): +5 feat bonus to death saving throws
- Eerie Resurgence (requires level 21 Revenant): +2 feat bonus to death saving throws, and you can spend an extra healing surge at a result of 20
- Elemental Guardian (requires level 11 Genasi Fighter with 15 Constitution): Increase earthsoul genasi's racial saving throw bonus by 2
- Human Perseverance (requires Human): +1 feat bonus to saving throws
- Imperishable Destiny (requires Human): +1 bonus to saving throws for each milestone you've reached
- Iron Resolve of Zerthadlun (requires Githerazi): +2 feat bonus to saving throws while you have at least 1 power point
- Psionic Fortune (requires Psionic Augmentation): +1 feat bonus to saving throws while you have at least 1 power point
- Resilient Focus: +2 feat bonus to saving throws
- Stubborn Survivor (requires Human): +2 bonus to saving throws while you have no action points
- Unyielding Fortitude (requires level 21): +2 feat bonus to death saving throws
- Crown Jewel of Chessenta (level 15 Rare): +2 item bonus to saving throws
- Emblem of Ossandrya (Paragon): +1 item bonus to saving throws
- Fortune Stones (set, requires at least 5): Each ally carrying a Fortune Stone gets +1 item bonus to saving throws
- Fragment of the Eleventh Rack (level 18+ Rare): +1 item bonus to saving throws
- Horreb Ritual Cube (level 18 Rare): +2 item bonus to saving throws
- Nullifying Ring (level 30 Uncommon): +3 item bonus to saving throws
- Ring of Protection (level 17 Rare): +1 item bonus to saving throws
- The Invulnerable Coat of Arnd (Heroic): +2 item bonus to saving throws
- Anointed Champion (requires Cleric) level 16 (Anointed Helm): +2 bonus to saving throws
- Scion of Absence (requires voidsoul Genasi) level 16: +2 bonus to saving throws
- Mul Battle Slave (requires Mul) level 11: +2 bonus to saving throws while bloodied
- Warforged Juggernaut (requires Warforged) level 16: +2 bonus to saving throws
So, let's throw together this hypothetical character.
Background: Scorned Noble; this will lets us become infallible for all saves if we back away from out comrades (or get them to back away from us!)
Total: +2 while further than 5sq from allies
Character Theme: The four choices don't matter as far as saving throw boosting is concerned.
Total: +1 always +2 while further than 5sq from allies (+3)
Race: Genasi, Earthsoul for the racial +1
Total: +2 always +2 while further than 5sq from allies (+4)
- Class: Fighter, so that we can pick up Elemental Guardian and boost the racial bonus. Multiclassing into Cleric for the paragon path
- Ability Scores: Wisdom at least 13, Constitution at least 12 are all that's needed at level 1 (assuming Constitution is raised at 4 and 8) to get the desired feats.
- Skills: Why not train Heal, to help your poor allies who aren't immune to save-ends effects and death?
- Powers: While there are powers which can boost saving throws, none last for longer than an encounter.
- Disciple of Death
- Born of Shadow
- Initiate of the Faith
- Elemental Guardian
Total: +4 always +2 while further than 5sq from allies +1 in dim light or darkness +5 vs. death saving throws (+12)
Paragon Path: Anointed Champion, selecting Anointed Helm
Total: +6 always +2 while further than 5sq from allies +1 in dim light or darkness +5 vs. death saving throws (+14)
Items: Nullifying Ring
Total: +9 always +2 while further than 5sq from allies +1 in dim light or darkness +5 vs. death saving throws (+17)
Now, a 3 on a death saving throw will let you spend a healing surge, and you can't fail the roll. You also can't fail a normal saving throw roll unless penalties are being applied to you (and those penalties have to be greater than -4 if the lighting isn't too hot and your allies are across the room).
For most actual games, a Nullifying Ring won't be available to a level 16 character, however. Even without any item bonus at all though you've got a healing surge on 6+ for a death save, and you still have 85% success rate for saves when your teammates are crowding too close and the lighting is good. And you can always add in one of the lower-level items which provide a bonus until you can get your mitts on the level 30 ring. The Fortune Stones set would be a nice lower-level choice, as each item offers a daily reroll for certain attack types, and once you get 5 of them everyone in the party is getting the +1 bonus.