First, let's simplify the expression inside the square root:

[ \sqrt{75 - 0.1 \sqrt{300 - 27} } ]

[ = \sqrt{75 - 0.1 \sqrt{273} } ]

Next, simplify the square root inside the square root:

[ = \sqrt{75 - 0.1 \times 3 \times \sqrt{91} } ]

[ = \sqrt{75 - 0.3 \sqrt{91} } ]

Now, the expression inside the square root is simplified. We can simplify it further:

[ = \sqrt{75} - \sqrt{0.3 \sqrt{91}} ]

[ = \sqrt{75} - \sqrt{0.3} \times \sqrt{\sqrt{91}} ]

[ = \sqrt{75} - \sqrt{0.3} \times \sqrt{91^{\frac{1}{2}}} ]

[ = 5\sqrt{3} - \sqrt{0.3} \times 9 ]

[ = 5\sqrt{3} - 3\sqrt{3} ]

[ = 2\sqrt{3} ]

Therefore, [ \sqrt{75 - 0.1 \sqrt{300 - 27} } = 2\sqrt{3} ]