How long can a RSA public key be?
You say DER, but you tag PGP which doesn't use DER -- and whose publickey block includes a lot of things beyond the RSA values (n,e). The DER encoding of a PKCS#1 RSA publickey of 4096 bits with a conventional e of 65537 (default at least for openssl and Java) is

30 82 02 0a   02 82 02 01 (n with leading 00)   02 03 01 00 01 


with a total length of 526 octets. The DER encoding of a generic SubjectPublicKeyInfo which is always used in X.509 certificates and often in other places (including openssl and Java) adds an outer SEQUENCE and AlgorithmIdentifier and a wrapping BIT STRING giving a total length of 550 octets. If you mean something else, specify it more clearly.

For the private exponent see RSA: Most significant key bit always 1? .

But if you are concerned about the encoding of a private key, that is much more than just the exponent and (possibly) modulus; see https://stackoverflow.com/questions/5403808/private-key-length-bytes . In addition the privatekey is often encrypted, which usually expands the encrypted value and adds metadata about the encryption. 

If you have a question about a particular key encoding with a particular program or protocol/format, you need be much more specific.