What would currently make the strongest https connection? when you look at the encryption websites use you will see that there is alot of diversity, some use 256 bit keys some do not, some use DHE-RSA some ECDHE some use sha 256 some sha1, but my question is, what would currently make the strongest https connection in terms of asymmetric  symmetric and hashing algorithms used.(I do understand that there might not really be an overall "strongest" one and that some schemes a better in different situations)
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