Solution:

Every option is odd, so only the primes $3, 5, 7$ need checking.

Option A: $2^{606} \equiv 1 \pmod 3,$ so $2^{606} - 1$ is divisible by $3.$ Option B: $2^{606} \equiv 4 \pmod 5,$ so $2^{606} + 1$ is divisible by $5.$ Option D: $2^{607} \equiv 2 \pmod 3,$ so $2^{607} + 1$ is divisible by $3.$ Option E: modulo $5,$ $2^{607} + 3^{607} \equiv 3 + 2 = 5 \equiv 0.$

For $2^{607} - 1:$ it is $\equiv 1 \pmod 3,$ $\equiv 2 \pmod 5,$ and (since $2^3 \equiv 1 \pmod 7$ and $607 \equiv 1 \pmod 3$) $\equiv 1 \pmod 7.$ So it is not divisible by any prime below $10.$

Thus, the correct answer is C.