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The Greek letter sigma is used for repeated addition. The Greek letter pi( uppercase, not lowercase), is used to denote repeated multiplication. In the same way as the last two, which symbol is used to show repeated exponentiation? (Notice: I already know about up-arrow notation but that's not what I meant.)

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  • $\begingroup$ Look up arrow notation for exponentiation (Knuth up-arrow). $\endgroup$ – D.B. May 21 at 19:12
  • $\begingroup$ There is no common usage. $\endgroup$ – copper.hat May 21 at 19:12
  • $\begingroup$ @D.B. If you are referring to this, note that the terms must be the same, unlike summation and product notation which allows the terms to be different. $\endgroup$ – angryavian May 21 at 19:15
  • $\begingroup$ I was not talking about up-arrow notation. $\endgroup$ – elipson_-1 May 21 at 19:16
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    $\begingroup$ It doesn't seem like any such notation would get any sort of frequent use regardless. For such an uncommon operation, defining it recursively seems fine. Generally, we only bother giving symbols to things which are useful. $\endgroup$ – JMoravitz May 21 at 19:17
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It's rare enough that it is apparently not worth the effort to invent, teach and learn a new piece of notation for it, the way we have with $\sum$ and $\prod$.

We do have special notation for when all the exponents are equal, the same way that if we add together many equal things we can write it as a product, as if we multiply together many equal things we can write it as a power. This is called tetration, and I have seen $5^{5^5}$ denoted both as $^35$ and as $5\uparrow\uparrow 3$. The latter is part of a hierarchy of operators known as Knuth's up-arrow notation, which is commonly used for writing numbers that are too large for regular exponent notation to adequately describe.

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