2. The first digit after the decimal point of $(1+\sqrt{2})^{2021}$ is $\qquad$ .
2. 9 .

Considering the conjugate $(\sqrt{2}-1)^{2021}$, it is easy to see that
$$
(1+\sqrt{2})^{2021}+(1-\sqrt{2})^{2021} \in \mathbf{Z} \text {, }
$$

and $0<(\sqrt{2}-1)^{2021}<1$.
From $(\sqrt{2}-1)^{4}<0.5^{4}=0.0625$, we get
$$
(\sqrt{2}-1)^{2021}<0.0625 \text {, }
$$

which means the first digit after the decimal point of $(\sqrt{2}-1)^{2021}$ is 0, hence the first digit after the decimal point of $(\sqrt{2}+1)^{2021}$ is 9.