Climbing Powers

$2\wedge 3\wedge 4$ could be $(2^3)^4$ or $2^{(3^4)}$. Does it make any difference? For both definitions, which is bigger: $r\wedge r\wedge r\wedge r\dots$ where the powers of $r$ go on for ever, or $(r^r)^r$, where $r$ is $\sqrt{2}$?
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Problem



We can define $2^{3^{4}}$ either as $(2^{3})^{4}$ or as $2^{(3^{4})}$ . Does it make any difference?

Now calculate $\left(\sqrt 2^{ \sqrt 2 }\right)^{ \sqrt 2 }$ and $\sqrt 2 ^{\left(\sqrt 2 ^{ \sqrt 2 }\right)}$ and answer the following question for the natural extension of both definitions.

Which number is the biggest \[ \sqrt 2 ^{\sqrt 2 ^{\sqrt 2 ^{\sqrt 2 ^{.^{.^{.}}}}}} \]

where the powers of root $2$ go on for ever, or \[ \left(\sqrt 2 ^{\sqrt 2 }\right)^{\sqrt 2} ? \]