Doubling a call's maturity from 100 to 200 days

You see the price of a European call with $100$ days to expiry. Roughly how should the price of the same call with $200$ days to expiry compare?

Higher, but well under double: roughly $\sqrt{2} \approx 1.4$ times for a near-the-money call. Option value is driven by $\sigma\sqrt{T}$ (the ATM approximation is $0.4\,\sigma S\sqrt{T}$), so doubling $T$ scales it by about $\sqrt 2$. Far from the money or with large rates the ratio differs, but it stays below $2$ for a non-dividend call.