Ta(7) is still not found yet. Maybe we will wait for the super mathematicians for few more years and they get the value of it.

So far, the ever-found biggest Taxicab number is 

$\text{Ta}(6)=\;24153319581254312065344\\ =582162^3 + 28906206^3\\ =3064173^3+ 28894803^3\\ = 8519281^3 + 28657487^3\\ =16218068^3 + 27093208^3\\ = 17492496^3 + 26590452^3\\ =18289922^3 + 26224366^3$

Enough mind-blowing isn't it?

~The smartest cookie in the world