Come on Badinage.  You have to tell us what tht equals!

Use the forum calc like I did   

Xerxes,  I suppose it depends on your take.   

Tell me, does 2 have a zero in it?   It is exactly the same as 2.0  so by your logic 2 must also have a zero.

Zeros are often necessary in numbers because they are place holders.  BUT 2 and .2 do not need any zeros as place holders because they are right up against the decimal point so their place value is indisputable.

And yes xy is x*y  so if you are not allowed to use a multiply sign, does that mean that you cannot multiply OR does it mean that you cannot put a * in the display.  I guess that is up to the individuals interpretation.  

I don't know but I will just start an answer thread here.  ( the calc puts the zeros in but the are not needed)

How about

$${{\mathtt{22}}}^{{\mathtt{0.22}}} = {\mathtt{1.973\: \!935\: \!646\: \!137\: \!177\: \!9}}$$

$${{\mathtt{2}}}^{{\mathtt{0.222}}} = {\mathtt{1.166\: \!349\: \!369\: \!788\: \!467\: \!3}}$$

$${{\mathtt{2}}}^{\left({\mathtt{0.2}}{\mathtt{\,\times\,}}\left({\mathtt{0.2}}{\mathtt{\,\times\,}}\left({\mathtt{0.2}}\right)\right)\right)} = {\mathtt{1.005\: \!560\: \!580\: \!398\: \!468\: \!2}}$$

$${{\mathtt{0.2}}}^{\left({\mathtt{0.2}}{\mathtt{\,\times\,}}\left({\mathtt{0.2}}{\mathtt{\,\times\,}}\left({\mathtt{0.2}}\right)\right)\right)} = {\mathtt{0.987\: \!207\: \!031\: \!388\: \!177\: \!8}}$$

$${{\left({\left({\mathtt{0.2}}{\mathtt{\,\times\,}}\left({\mathtt{0.2}}\right)\right)}^{{\mathtt{0.2}}}\right)}}^{{\mathtt{0.2}}} = {\mathtt{0.879\: \!189\: \!311\: \!540\: \!889\: \!8}}$$

Well my answers are getting smaller     

I think you can not take the number 0.22, as the 0 itself is a digit, isn't it? And the 0 is not a 2, Even Rough You don't need write it  

But correct me if I am wrong 

I did not write it the calc put it in there 0.2 = .2     the 0 is not needed.

 It is usually added just so that the . is not mistaken for a speck of dirt.

I know 

But the number itself still contains the digit 0, doesn't it? 

xy contains a times sign as well, doesn't it? Even tough we don't have to write it?

$$\textcolor[rgb]{\mathbf{0,0,1}{.2^{2^{22}}}}$$$$\textcolor[rgb]{0,0,1}{\mathbf{.2^{2^{22}}}}$$

  evaluates to ∼ 2x10^-2931693

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I think your answer wins Badinage         Your answer puts mine to shame.    LOL 

I think Badinage's answer is great, I am now able to accept the .2 as a proper number. Without the 0. This nice construction is quite fascinating 😁

I am glad that you are happy xerxes, Badinage's answer is great and playing with numbers is always fund for mathematicians and mathematicans in the making :)

It is indeed, and you learn a lot by doing this. Mathematicians in the making is a nice word, btw.

Melody> And yes xy is x*y  so if you are not allowed to use a multiply sign

But you are allowed to use brackets, so it's permissible to write x*y as x(y)

Or you could just writie it as xy

Either way there is an invisable * sign