The value of $$\sqrt{2023\sqrt{2022\sqrt{2021 \times 2019 + 1} + 1} + 1}$$ is

The innermost quantity is $$2021 \times 2019 + 1$$ which can be written as $$(2020+1)(2020-1) + 1$$

Using the property that $$(a+b)(a-b) = a^2 - b^2$$ we get 

$$(2020+1)(2020-1) + 1 = 2020^2 -1^2 +1=2020^2$$

Hence, the square root of the innermost quantity is $$2020$$. The next radical becomes $$\sqrt{2022 \times 2020 + 1} $$

We can again rewrite this using the same property mentioned above to get $$(2021+1)(2021-1) + 1 = 2021^2-1^2 +1 = 2021^2$$

$$\sqrt{2021^2} = 2021$$. Now we can again rewrite this in the same way to get 

$$\sqrt{2023 \times 2021 + 1} = \sqrt{(2022+1)(2022-1) +1} = \sqrt{2022^2-1^2+1}=2022$$.