A little comparison

Consider the ratio:

$\begin{aligned} \frac {222^{333}}{333^{222}} & = \frac {2^{333}\times 111^{333}}{3^{222}\times 111^{222}} = \frac {8^{111}\times 111^{111}}{9^{111}} = \left(\frac {888}9\right)^{111} > 1 \end{aligned}$

$\implies \boxed{222^{333} > 333^{222}}$

First consider $222^3 \text{ and } 333^2$

$222^3>200^3=8\cdot 10^6$

$333^2<400^2=16\cdot 10^4$

Hence

$222^3>8\cdot 10^6>16\cdot 10^4>333^2$

$\Longrightarrow 222^3>333^2$

Raise both sides to the $111^{\text{th}}$ power

$(222^3)^{111}>(333^2)^{111}$

$\Longrightarrow \boxed{222^{333}>333^{222}}$

222^333/333^222 = (2 111)^333/(3 111)^222 = 2^3^111 * 111^(333-222) / 3^2^111 = 8^111 * 111^111/9^111 = (888/9)^111 = (98.6...)^111 >>>>1 ; 222^333 > 333^222

$222^{333} = 222^{222} × 222^{111} > 222^{222} × 2.25^{111} = 222^{222} × (1.5^2)^{111} = 222^{222} × 1.5^{222} = (222 ×1.5)^{222} = 333^{222}$

Hence:

$\boxed { 222^{333} > 333^{222} }$