Probably the single best ice in Standard at the time of writing this. Logjam is a gamewinning ice that completely hoses several decks if they don't run silver bullets, shuts down several of the most common cards in the format, and performs well against everything else. Even in it's worst showing it is pretty much a strictly superior Pharos, though pharos doesn't get run and logjam is a 3 of in every BtL for a reason.
As a facecheck:
Naturally, this is logjam's worst performance. Logjam is 6 to rez, and refunds 2, for a net cost of 4 and no cost to the runner.
As regular ice:
Logjam with a single advancement counter is 6 to rez, and has a strength of 4-6 (asset + operation are mostly guaranteed, ice + upgrade are common but may not be in your archives earlier in the game). With 3 subroutines which must all be broken, this makes it at worst about par and at best 2 strength better than the average. For comparison, pharos is -1 to 1 more/less strength for an extra credit, and the runner doesn't have to break one of the subroutines if they don't want to.
Against common scams:
Boomerang is completely shut down without any recourse against logjam. At best they can burn it to deny you the 2cr or recover it on a later run when they are able to break. This is really phenomenal and lets you play cards behind a single logjam with a confidence that you don't normally have against hoshiko or shapers.
Tsakhia "Bankhar" Gantulga pays three subroutines to get past logjam. Nothing special, but gives a more or less 1:1 return on investment.
Botulus can't break this on the turn that it is played, even with Cookbook. This makes Simulchip botulus much less threatening. It also takes three virus counters to break logjam in general, reducing how often they can run and giving an actual use to botulus'd ice.
Audrey v2 needs two virus counters to break logjam's three EtRs, which makes virus counter management much easier in slow grindy games against freedom and the like. Logjam has no special effectiveness against Aumakua other than just being really efficient and high strength ice.
Against actual breakers:
Cleaver is the most common fracter in standard, and is run by most anarchs and most shapers. Other than cleaver, you will mostly see AI, Propeller and curupira. Logjam absolutely eats cleaver for breakfast: every advancement counter is another 2cr the runner has to spend on any run past logjam. Double turbine cleaver pays 12cr to break 12 str logjam, and the cost just goes up.
Unlike pharos, logjam actually beats turbine Propeller. Propeller can't break logjam in the lategame when it is advanced past 8/10 str, and the runner is forced to switch back to cleaver - by which point, breaking with cleaver has a cost well into the double digits.
Curupira only treats logjam as any other large and efficient barrier. It becomes important not to have cheap barriers on the outside of servers to prevent the runner from bypassing logjam by farming charge against these, but for the most part this is just something you have to deal with and conventional criminal breaker suites have the best time against logjam in the current meta.
Special interactions:
Unlike a lot of large and expensive ice, Logjam gets it's money back in a way when it is derezzed - more counters will get added to it when it is rezzed again. This means that if a runner actually wants to pass logjam, then logjam is still cost effective to rez while it's under a Tranquilizer.
Logjam gets forcibly set to 0 strength under Hush. It's a testament to how strong logjam is as a piece of ice that Lat now runs hush specifically to deal with logjam/treeline.
As you mention here, it seems likely this card exists to be a repository of advancement counters that you don’t have to spend clicks on rather than a particularly strong ice. As a result I love the design here - a role player in certain decks where the synergy with advancement counter trickery will be more useful than the better stats of Pharos.
— Ksym777