What would currently make the strongest https connection?
The cipher suite with the best theoretical security for TLS (v1.2) right now is clearly:

TLS_ECDHE_ECDSA_WITH_AES_256_CBC_SHA384 with P-521
as the associated curve for the certificate and the key-exchange.
The arguments are simple: 


ECDHE and ECDSA with P-521 both provide 256-bit security against forgery / key recovery attacks
AES-256 is the strongest possible AES instance you can get
CBC is the most resilient mode to IV-reuse (you only learn whether two messages share a common prefix)
SHA-384 implies use of HMAC-SHA-384 as the MAC which has a much higher "bit-length-protection" against forgeries than, say GCM


The arguments against it are clear as well:


The support situation is probably horrible, because P-521 isn't exactly the most wide-spread supported curve as is probably the ECDSA-CBC-SHA2 combination
This suite is prone to many (side-channel / padding oracle) attacks on CBC and authenticate-then-encrypt from the past


Note well: Because of these attacks I strongly recommend against using any CBC based cipher suite.

The cipher suite with the best practical security (and somewhat decent support) for TLS (v1.2) right now is clearly:

TLS_ECDHE_ECDSA_WITH_AES_256_GCM_SHA384 with P-384
Downgrade to TLS_ECDHE_WITH_AES_128_GCM_SHA256 with P-256 / P-384 for more supported systems (such as Firefox)


ECDHE and ECDSA with P-384 both provide 192-bit security against forgery / key recovery attacks, which is on the edge of "physically infeasible" and will likely remain unbroken until the rise of quantum computers, the support is really good here, because P-384 is part of NSA's Suite B and many people implemented it because of that
AES-256 is the strongest possible AES instance you can get
GCM is the mode which has seen the least amount of attacks, as predicable IVs are a non-issue and padding doesn't need to take place and authentication is built-in
GCM "only" provides 128-bit forgery security, but this is (a worthy / the) trade-off for a much better side-channel resistance