Two

Two is a language invented by User:Tailcalled. It is an uncomputable extension to a Brainfuck dialect, designed to increase the power as much as possible with a single instruction.

Two has nine instructions, eight of which are similar to Brainfuck. The syntax rules are the same as in Brainfuck.

Instruction might seem a bit vague, but can be explained as follows: we say that the effect of a program is its I/O behaviour and whether it terminates. Thus, the effect of the program + is "nothing"; a program consisting of + simply terminates. Instruction will then increase the value pointed to until further increases do not change the effect. If that cannot be done (for example, in the program I[->.<]), Instruction will loop forever.

Instruction has been inspired to adding an "infinite" number to arithmetic by exploiting the compactness theorem.

Computational class
Two is easily able to solve the Halting problem, so it is more powerful than Turing-equivalent languages. The exact power of Two is unknown, but it would not be unreasonable to question its consistency or well-definedness.