Composition

Primes, primes, every where,
Was all the bard did think;
Primes, primes, every where,
But nary one in link.*

This base-ten sequence exhibits an absence of prime linked primes (that is, the concatenation of any number of consecutive terms) in an infinite sea of primes:

2, 5, 11, 13, 29, 31, 17, 19, 43, 7, 37, 41, 71, 47, 67, 89, 3, 101, 23, 109, 59, 83, 103, 73, 107, 157, 53, 127, 149, 61, 131, 139, 79, 163, 191, 193, 97, 113, 137, 167, 211, 181, ...

Such sequences are not rare, this one being the lexicographically first. Here is the base-two analogue:

2, 5, 17, 13, 11, 23, 3, 19, 7, 53, 37, 31, 47, 29, 43, 59, 41, 73, 67, 83, 89, 61, 79, 71, 107, 97, 127, 131, 101, 113, 151, 103, 137, 109, 167, 179, 139, 227, 149, 191, 157, 193, ...

Here is one that works in either base-two or base-ten:

2, 5, 17, 43, 7, 23, 19, 127, 11, 41, 157, 101, 13, 131, 3, 211, 37, 149, 163, 173, 31, 107, 229, 29, 89, 67, 109, 223, 73, 193, 47, 79, 59, 71, 179, 191, 151, 97, 269, 139, 277, 227, ...

And this one works in any base from two to ten:

2, 229, 131, 263, 37, 421, 491, 223, 911, 127, 167, 383, 1187, 401, 31, 15307, 701, 971, 2797, 3, 8741, 571, 5477, 6037, 619, 859, 6359, 353, 2659, 311, 3851, 379, 7193, 7993, 3319, 653, 691, 13441, 661, 1579, 7541, 1987, ...

* Primes of the Ancient Mariner