Dilute ammonia does the trick for me.  This takes the itch away, some say, because it is a base and the itch irritant or itch response requires an acidic environment. 

I can testify from personal experience (!) that this leaves less scarring than from constantly scratching these bites, or from spraying freeze mist on them, both of which I tried in desperation one night after a day hiking in the rainforest.  Don't do either of those things...but hiking in the rainforest is nice.

Hm... I certainly agree that the worst treatment is scratching. Not only is it damaging, but the damaged tissue soon begins to itch even worse than before and you are into a vicious circle before you know it. If you have nothing else to use, try slapping the spot; it works about as well as scratching, and causes less damage.

Note that everything I say ignores any requirement to deal with any infections resulting from the bite and similar problems. There are many such, depending on what bit you and where in the world you happened to be at the time.

Anyone who has experienced really bad itching in reaction to bites from midges, mosquitoes, sandflies and the like, will curse my next bit of advice as they feel insanity closing in, but it works better than anything else I know for mosquitoes, ticks and ceratopogonids at least. (Cleg bites that I know don't itch much.) Possibly some antihistamines would do better for some bites, but only if they are applied early, before the tissue damage has set in. And possibly some local anaesthetics would help.

The advice? Do as little as possible. Protect the spots from all forms of scratching and other interference. A bit of softening skin lotion could do no harm. 

Itching??? Never heard of it.  I am a rock! (At least from the neck up!) The thing is that in the absence of infection or bits of the biting organism broken off inside your skin, the itching begins to subside in a couple of days. Scratch or apply the wrong lotions and oils, and you still will be in purgatory a fortnight later, and scarred for much longer.

Also, if you can get used to not scratching, the itch becomes something subliminal. You can go without noticing it for hours on end. If you scratch you are acutely aware of it from minute to minute, which not only is torture, but torture refined with frustration and boredom (just try a few hours obsession with one itch and see how boring it becomes! Especially to your associates!)

So do as you please, but avoid tissue damage, grit your teeth and find something else to concentrate on. (Two more days... Two more days...)