$2^{101}$

Observe the following sequence :

$\begin{aligned} 2^1 = \color{#D61F06}2 \\ 2^2 = \color{#3D99F6}4 \\ 2^3 = \color{#20A900}8 \\ 2^4 = 1\color{#EC7300}6 \\ 2^5 = 3\color{#D61F06}2 \\ 2^6 = 6\color{#3D99F6}4 \\ \end{aligned}$

If you observe the end digits of every term follow the sequence $\color{#D61F06}2$ , $\color{#3D99F6}4$ , $\color{#20A900}8$ , $\color{#EC7300}6$ . So, the end digit of $2^n$ is $\color{#EC7300}6$ if $n$ is a multiple of 4. So, the end digits of $2^n$ where $n$ is $4, 8, 12, 16, 20, ..... 100, 104, ...$ will be $\color{#EC7300}6$ .

If the end digit of $2^{100}$ is $\color{#EC7300}6$ then naturally the end digit of $2^{101}$ is $\color{#D61F06}2$

$\begin{aligned} 2^{101} & \equiv 2^{4\times 25+1} \text{ (mod 10)} \\ & \equiv 16^{25} \times 2 \text{ (mod 10)} \\ & \equiv {\color{#3D99F6}6}^{25} \times 2 \text{ (mod 10)} & \small \color{#3D99F6} \text{Powers of 6 always end with 6.} \\ & \equiv 12 \equiv \boxed{2} \text{ (mod 10)} \end{aligned}$