Leading digits of a power

Let $e_n$ the sequence of integers such as the decimal development of $2^{e_n}$ starts with the following digits: $66666\dots$

For instance we have:
$e_1=224296\to 2^{224296}=66666468242741993302\dots$
$e_5=620220\to 2^{620220}=66666969606590729636\dots$

What is $e_{1000000}$?


[My timing: 40 sec]