What about the P-RAM battery? Did you exchange it already?
Rebuild the desktop file, that may help. TechTool was pretty good at fixing things.
http://macintoshgarden.org/apps/techtool
Boot Norton Utilities from your CD-ROM and let DiskDoctor and SpeedDisk do their jobs.
http://macintoshgarden.org/apps/norton-utilities-353-0

LCIII's and some other 68k Macs are notoriously difficult to boot up if the PRAM battery is dead. Removing would not help, if your problem is battery related. Replace it if the voltage has dropped below 3 volts (should be 3.6 preferably).

I usually order my stuff here:
http://www.reichelt.de/Lithium-Batterien/TEKCELL-AA02/index.html?;ACTION...
but you have to shop for € 10,- as a minimum.

I don't know if a dead battery would necessarily stop a LC III from starting up, but I had mine running fine with no battery at all.

I'd be more inclined to look at the capacitors on the logic board or psu. My Classic II had that occasional wait before chiming and then sometimes vertical stripes. Eventually it wouldn't go any further. I washed the board in distilled water, left it to dry for a couple of days and it boots fine now. I don't know how much time a logic board wash will buy you before you have to replace the capacitors, but it's one thing to consider.

You could try to connect a external 4,5V power source to your battery´s connectors on the board.
If that would not improve things, you might save the money for the Li Battery.
If your logic board is really filthy, washing it might help.
If ELCO´s start leaking, the corrosive electolytics will ruin the board sooner or later and the caps wont work as specified.

The batteries are just a few $ each if you get them from the Stotes, OWC (macsales.com) is where I get mine. Be prepared to pay for postage, so it only gets cheap if you buy 6 or so, handy if you have a small collection of pre-Intel Macs. Can't remember how many I bought last time, I only have one left - handy if one of the NZ members needs it in a hurry I suppose.

@dougal: If you're in the US you can find a replacement battery for approx. $7 shipped. See "The Upgrade Box" on this Low End Mac page.

@bertyboy: When I was last in NZ I had to find a replacement 3.6v battery. I found it quite a challenge finding one locally. I then discovered Jaycar Electronics stocks and sells them in bulk. NZ$14.50 for a single item, which isn't too bad considering currency conversion to Sterling or even an Aussie $. Good if you have a Jaycar store handy and you're in a hurry

Catalog # sb1770 on the jaycar.co.nz site

MikeTomTom, thanks for that, added to my favourites. There's about 5 in Auckland, including one in Newmarket, which I'll pass everyday to and from the office. For a cosmopolitain shopping area, it has a lot of Mac friendly shops, it also has the Apple shop.
All my Macs have new batteries, so I wasn't planning on needing any more, but I found myself searching through trademe and trade & exchange last night looking for a B&W G3 tower (none), a 20in Cinema Display ($20), an old AppleTV (more than the new one, new), and a selction of early G4 desktops (too expensive), so I may need more soon.

@bertyboy: Ah Newmarket, hub of the (Auckland) universe - My favourite strip to drive through at 5 PM on a week night

There is one G3 tower on trademe, no bids, start price $30 and ending this Friday (its located in Auckland too). - Listing # 382322771

I want one of these myself. Love the look of them and haven't owned one as yet. I went from beige G3 to a G4 tower.

doh, how did i miss it last night. but $30 is too much. I bought one in the Uk for £0.99 ($2) but sold it before I left. Shame.

LCIII: clicks - a dead hard disk ? but that's more like a rattle. the clicks used to mean a dead power supply, but if you're getting power anywhere it's not that.