jmc

We have $$\begin{array}{rl}66666_{16}& =6\cdot 16^4+6\cdot 16^3+6\cdot 16^2+6\cdot 16+6\\ & =6\cdot (16^4+16^3+16^2+16+1)\\ & =6\cdot (2^{16}+2^{12}+2^8+2^4+1)\\ & =(2^2+2)\cdot (2^{16}+2^{12}+2^8+2^4+1)\\ & =2^{18}+2^{17}+2^{14}+2^{13}+2^{10}+2^9+2^6+2^5+2^2+2.\end{array}$$Actually, this is more detail than necessary; what is important is that $2^{18}\le 66666_{16}<2^{19}$, which tells us that the base-2 expression of this number has $\boxed{19}$ digits or bits (with place values $2^{18},2^{17},2^{16},\dots ,2^2,2^1,2^0$).