Can you make sourdough starter without throwing any away?
I just asked the exact same question here!:
Why throw away so much sourdough starter? Tartine Book no. 3

I found a pretty good answer online. It actually would be easier and cheaper for you to throw it out, as you don't have to buy as much organic flour in the future to feed your starter. Imagine buying 5 cups each day to feed your starter when you could just use 100 grams!:


  During the "initial" development - and by this I assume we're talking
  the first several days of your brand spanking new starter, what you'll
  be throwing away is likely not even "starter" but rather flour soup
  wherein there's a fairly good chance you're also growing a few
  less-than-desirable cohabitants. This would explain why you'd want to
  chuck it out rather than use it.  You also cut back on your starter so
  you don't end up with gallons of it since quite early on you do begin
  to make your feeds at a ratio of 1:1:1 or 1:2:2.  For illustrative
  purposes, let's follow the 1:2:2 ratio and just assume we're using
  cups (easier to visualize) although you really would be weighing, of
  course. So let's say  on day 1 morning feed, we begin with a half cup
  of starter. Half a cup sounds like nearly nothing, right? To that we'd
  then need to add twice that amount of each water and flour, so we add
  1 cup water and 1 cup flour for a total of 2.5 cups. Still not that
  bad. Day 1 evening feed, we're not reducing so we need to add to our
  2.5 cups of starter 5 cups water and 5 cups flour for a total of 11 cups. Set it on the counter, hope it doesn't overflow everywhere and
  go to bed. Day 2 morning feed, we're still not tossing away so to our
  11 cups we have to add 22 cups of water and 22 cups of flour. We now
  have 55 cups of starter or about 3.5 gallons. Remember, this is only
  the third feed and we started with just a measly half cup yesterday
  morning. At tonight's feed, we'll have 312 cups or 19.5 gallons of
  starter. Just enough to fill up four of those white Home Depot buckets
  although you'll want at least eight of 'em so the starter can double
  in size. By morning of day three, you can fill your bathtub, if it
  holds 97 gallons; it very well may not, and definitely won't hold the
  expanded starter. By afternoon, you'll also be borrowing an additional
  four neighbour's tubs for your 488 gallons.  How soon before you need
  to move this out to a local swimming pool to continue? Well, by the
  end of day five, you'd be the proud owner of 305,175.8 gallons of
  starter. Or 4.8 million cups. Seriously. An Olympic size pool is about
  600,000 gallons. By the morning of day six, you can fill two and a
  half of them, with leftovers. And it hasn't even been a full week yet.
  And that's why you cut your starter back down to a manageable amount
  each time. It's a lot cheaper to toss a little bit out than to keep it
  and feed it.


Source: http://www.thefreshloaf.com/node/12467/throw-away-half