Cryptic crossword clue
Old elixir, consumed at nine perhaps, for increase in power 14

The most common and most likely answer to this crossword clue is the 14 letter word EXPONENTIATION

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Buddy explains!

The straight part
We know that "increase in power" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case increase in power = exponentiation.
The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.
This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.

What does EXPONENTIATION mean?

EXPONENTIATION noun
  1. The process of calculating a power by multiplying together a number of equal factors, where the exponent specifies the number of factors to multiply.
  2. A mathematical problem involving exponentiation.

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