I hope your best card is Psi-Blocker, because boy-oh-boy have we got some spicy memes for this one!
Unfortunately, the combo I'm going to highlight here does require the use of several cards that are banned from competitive play. However, if performed, the player would be utterly untouchable.
Back in 2014, a combo was found which, at the time, would take 140 days to complete... but would literally render every card in the game unusable until the player's next turn by repeatedly summoning and using the effect of Psi-Blocker until every card in the game (not including those required for the loop, of course) was declared. Now, let me reiterate... at the time this would take 140 days to complete. The combo is one of my favorites in card games, because over time, more cards are released - meaning that over time, this combo will take even longer.
The jist of the loop involves using the infernities, lavalval chain, hundred-eyes dragon, and a whole host of other infamous looping staples to summon Phantom of Chaos, banish Psi-Blocker, use its effect, summon Leviair the Sea Dragon, summon back Psi-Blocker, use its effect again, do some work to summon Dewloren, return your whole field to hand and do it all again.
More Along the Lines of What You Asked For:
A scenario where a card cannot be removed or countered by any means is difficult to imagine.. but I believe I have one.
EDIT: Updated after evilReiko pointed out a break. I believe it's now unbreakable, but please feel free to stress test it.
With Majesty's Fiend,
Imperial Iron Wall,
and Mask of Restrict
already in play, summon The Last Warrior from Another Planet.
With this field,
Your opponent cannot summon Monsters by any means (Last Warrior)
Your opponent cannot Set or activate any Spell/Trap cards, nor Set any Monster cards (Dark Simorgh + Anti-Spell Fragrance)
Your opponent cannot activate any of their Spell/Trap cards that are already Set (Denko Sekka)
Your opponent cannot activate Trap Cards like Infinite Impermanence, which activate directly from the hand (Jinzo)
Your opponent cannot activate any card effect in their GY by banishing the card (Imperial Iron Wall)
Your opponent cannot activate monster effects (Majesty's Fiend)
It is not possible for your opponent to take any action with the given board state, so none of your cards can be removed by any means.
I am aware that some might not consider this "making a monster indestructible." I am working on finding some set of cards that cannot be removed even while the opponent is able to do things, and will update this answer if I find one, but such a combination is extraordinarily difficult (if not outright impossible) to find.