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INTERCAL by Donald Woods and James Lyon

Compiler Language With No Pronounceable Acronym.  (Yes, that's what the name officially stands for!)  Commonly regarded as the world's first esoteric language, though Kvikkalkul is rumoured to predate it.  Designed one morning in May 1972 to be Turing complete while having nothing in common with any other language (though that does seem to be a contradiction).  The canonical example statement is the simplest way to assign a value of 65536 to a 32-bit variable:

DO :1 <- #0$#256

Though it claims to be capable of anything that any other language can do, the original version of the language restricts acceptable input to numbers with the digits spelled out, and output to an extended version of Roman numerals.

See also: C-INTERCAL, TriINTERCAL, Threaded INTERCAL

Iota by Chris Barker

A language created on the principle of Unlambda, with only a single combinator, namely λX . X s k where s and k are the combinators from SKI-combinator calculus.

Although the language has no input/output capabilities, Iota is essentially Turing-complete.