What size wire gauge for 1000ft service run?
Traditional, aka "nobody ever got fired for recommending this"

1000 feet underground, split phase 120/240V service, 3 wires, this is the main service to the house (so we get a favorable 83% derate on NEC 310.15b7, so calculating for 83 amps).   Your choices are 

  250 kcmil aluminum at 6.84% voltage drop,  $3,300
  400 kcmil aluminum at 4.88% voltage drop,  $5,400
 1000 kcmil aluminum at 2.80% voltage drop, $11,000
   3/0 AWG  copper   at 6.06% voltage drop,  $9,000
  250 kcmil copper   at 4.48% voltage drop, $12,000
  500 kcmil copper   at 2.86% voltage drop, $24,000


Some people are really hung up on 3% voltage drop.  Other people are really hung up on using aluminum nowhere ever, and ignoring the actual facts.  Start a Kickstarter and let those people pay the cost differential!

All these wires will also accept 200A service.  However voltage drop will be higher.  Voltage drop is proportional to actual current being flowed at the moment and has nothing to do with nameplate ratings.  

Transform the problem

Now, we can ease the pain by stepping up the voltage for the long haul.  We will use two transformers back to back.  These transformers are about $1,000 each for ones ample to your current draw.  

In between the transformers, voltage is stepped up, which means current steps down by the same proportion, so volts x amps still yields the same.  If we double from 240V to 480V, current halves from 100A to 50A.  240V to 600V (the max for common wires) drops current to 40A. We still get to derate 83%, giving 32.75 amps.  So... 

   4 AWG aluminum at 5.05% voltage drop, $  500
   6 AWG copper   at 4.89% voltage drop, $1,100
   1 AWG aluminum at 2.68% voltage drop, $1,000
   3 AWG copper   at 2.60% voltage drop, $2,000


WOW, that savings on wire is huge, and way more than pays for the transformers in all cases.  Here, it seems like the winner is #1 aluminum if you intend to fit 200A service in the future, otherwise #4 Al.  

I wasn't able to find any wires in the 6-7% voltage drop range, because the next smaller wire was giving 8-10% voltage drop. 

This is priced based on only 2 wires.  There is no point running a ground wire 1000 feet, even if this wasn't a double-isolated, doubly separately derived service, which it certainly is.   There is no value whatsoever to grounding your house to a point 1000 feet away.  This will be its own full and proper service, and it will absolutely depend on local grounding at the house, but that was already required. 

Theft

There are two ways to go: direct burial, or conduit.  Direct burial must be trenched 27" to allow 24" cover.  Conduit can be covered 18" (plastic or EMT) or merely 6" (rigid or IMC conduit). 

If you work in conduit, it is a foregone conclusion that some hick will break into your remote service panel, cut your leads, tie 'em to his truck's ball hitch, drive away with your 1000’, and spend the month trading chunks of it for meth.  Or worse, just cut the wire in a failed attempt to do that, and leave you 6 inches short on a 1000' run.  

For this reason, I would definitely bring the conduit to a wiring trunk a short distance from the house and remote site, bind the wire in some way so it can't be pulled out (or so the short bit pulls out), and then bury and hide the trunk.  This also helps you if your distance proves to be a little more than 1000', position the trunks a little less than 1000' apart so you can just buy full spools.