Hi!
First, thanx for the feedback. I'm glad that some people have found in my game log not only an entertaining story but also a sort of strategic manual on how to perform on max difficulty levels. I suggest you to continue following it. I have played 2 more turns but didn't update the log yet (i'll do it this evening). You probably will find a few more useful hints on the upcoming events.
Now, believe me or not, this is only the sixth deity game i've been played so far, not counting the few times i restarted immediately because of bad starts. In the first i lose culturally. In the second i've been utterly crushed. In the third, another cultural defeat. In the fourth, victory by space race. Fifth, won by domination. The sixth, well you know.
You ask how to deal with the initial tech race. You are probably wondering why the AS always seems to surge away in techs so fast at the beginning. Well, there is no discount on research involved. The reason is only the set bonuses given to them. They benefits of an extra settler, 2 extra workers and a whole bunch of military units.
The extra settler and workers allow the AS to found a very early 2nd settlement and to road tiles faster. And this means more gold available to be poured into research. With the free units, in most cases they manage to grab all the goody huts available before you have a chance to see them, and some of them contains techs. Then, they meet each other almost immediately, supposing they have a way to meet, so they can trade their starting techs each other. In deity and above, the AS trade between them like crazy with a surprising price-happy attitude. They form a sort of "discount club" in which you are not included. That's why you are robbed to death everytime you try to strike a deal with those bastards.
What to do then? You've already received good advice, but everything depends from the situation you're on, the civ you choose and even the version you're playing. I suspect that most of people here play Conquests. If you're using Vanilla or PTW, the rush from (Alphabet) Writing to Republic is less effective, since Philosophy don't award you with a free tech if you're the first to discover it.
The max research on Pottery may be a good option if you are producing some gold, but it may reveal itself a tremendous waste if, for example, you meet a civ that have Pottery as a starting tech when you're short 1 turn to finish researching it...
If you know Alphabet from the start, researching Writing at minimum is usually the best option. Writing is not known by anyone and you have good chances to be the 1st to discover it (and this means good trades). Then you may go for Literature, with the intention to build the Great Library (but you need masonry early to prebuild with the Palace) or simply because it's a tech that the AS tend to overlook (more good trades).
Another possible path to follow is the rush to Monarchy. If you know from the start warrior code of ceremonial burial, and if you're the Aztecs you know both, you can try max research on Mysticism, then go for Polytheism and finally grab Monarchy.
In definitive, there is no "best" option everywhere and everytime. Judge alone and figure out things by yourself whenever you can, but don't forget that in these forums you'll find tons of good advice for any situation. Good luck!
