What tech do you NEVER research? What ship equipment do you never use?

In a nutshell, I skip nothing.

Military Support techs give ships the abilities to buff your fleets and debuff enemy fleets for extra damage/range.

I think a Life Support System is probably cheaper than a Starbase, since they have maintenance now. Depends on Map size too I guess.

i play with reduced range on my maps so life support is a must at least for trade routes and such
i also turtle a lot so i use a lot of starbases

the techs i skip the most are the specializations that reduce maintenance and construction costs. if i had to choose between those two i would get the maintenance and skip construction cost

I usually try to get the tech that enables tourism as early as possible, but rarely research any tourism specific techs.  Like the original poster, I also don't usually research life support, although I have had one or two games where I've gone partway down that tree.  And since I'm usually not wanting for money, I don't research the cost reducing specialization techs either.

Tourism influence

I have yet to research the defense augments specific to each defense, just some overall evasion and jamming.  I have had reason or excuse to at least enter each of the other tech branches at least once or twice.  This is over a rather large number of hours and games.

I prioritize population, (food and growth) and production (Hives and governance) and research, research, research.  I delay military techs recklessly and it causes me great troubles mid game.  However, it keeps the game interesting. 

Funny I never research or get growth. All my colonies seem to max out pretty soon without it, even with multiple farms.

I never do growth, or happiness and rarely ever finish trade or income. I also dont bother with life support after 'Life Support'. Also the soldering line is not needed if you grab one of the first tier invasion tactics like biowarfare which is my invasion tactic of choice. Also the defense and hardening hit point regen line is last as I would rather just build a new ship than waste cargo space on a module that gives me 2 hit points per battle and my ship has 700+ hps. 

I find it interesting that others do not focus on growth, which to me is an obvious objective.  I don't say that my playstyle is superior.  It does start out bottom of the power graphs and spike dramatically late game, and that would never fly in a cut-throat MP game at all.  In the description of the Dread Lords it says if you leave them alone too long they become a big danger.  For the AI, that is true about me.  If they don't take me out early game, they are in great danger.  Often, when they have started their invasions a little too late, and I am building my fleet as they wreak their wanton destruction, my faction has the motto, "We may be in trouble, but you are doomed."

I consider it evidence that the devs have managed to design in different ways to play the game successfully, for different definitions of "successfully". That is a necessary and neat trick to pull off in a sandbox game, and it deserves mention and celebration.

It depends what map sizes you are playing on. If you are on a big map colonizing new planets every turn you may need growth. I find on a medium sized maps my planets have hit max pop already by the time I've researched my first few techs.

I tend to play on excessive and above sized maps only and play for extended periods of time so pretty much everything gets research or traded for eventually.