In standard arithmetic, 1+1=2 is a foundational truth established from the Peano axioms and proven rigorously — most famously across 362 pages of Whitehead and Russell's Principia Mathematica. No credible mathematical framework produces a sum of 3 from adding 1 and 1, and no peer-reviewed mathematician has ever claimed otherwise.
The apparent "exceptions" people encounter online are deliberate trick proofs. As documented in widely viewed YouTube breakdowns of these tricks, they all smuggle in an illegal operation — most commonly dividing both sides of an equation by zero — which makes any result possible and proves nothing. These are presented as puzzles or illusions, not valid mathematics.
The cultural shorthand "1+1=3" (meaning a couple can produce a child, or two ideas combine into something greater) is a figure of speech about synergy, not a mathematical identity. Sources like BYJU'S and GeeksforGeeks define addition unambiguously: combining two units of 1 yields a sum of 2, with no exceptions.