Doctor, if you aren't willing to spend money that pretty much narrows your options. I can't think of any completely free web hosting options at this second other than applying for a virtual domain, and then running a web server off your PC which is not the most ideal situation. Still that is possible to do and isn't very expensive.

Your best bet would be to get a domain and use Dropbox LOL or one of the other free storage hosts like Mega (50 GB for free).
What you could do is root the domain to your shared link and people could visit your shared link and see your website through that.
There are also some free website places you can find if you google free website, but they also have a need for a domain.
The cheapest thing you could do and still be really doing real web hosting is use amazon s3.
They are $0.03 a month for 1 GB. Every GB above that is also $0.03 till you reach 1 TB, then it goes down.
You still need a domain though.

There are a couple options that come to mind:
First, if you're willing to fiddle around, there are google sites. They have a bit of stuff at the top of each page, but they're workable.
Secondly, and I would personally say much better, is to just use dropbox. You're site will have a wierd link, but there's nothing easier than simply editing all of your files on your computer and having them instantly up and ready to go. Just put your site in your public folder, and give the link to index.html and you're good to go. The only problem with this approach is that you get slightly limited bandwidth (20gb/day for free, 200gb/day for paid).

My ishue with 000webhost, is I believe you have to fill out a captcha? I don't know, as I have not used it in like, 2 years.

Hey, Can you give that link again please? It doesn't seem to work...
Thanks,

www.free-webhosts.com, if it's still up, iss a complete directory of free hosts, complete with reviews and ratings, and detailed information about each free webhost.
  Or, if you want to go the dropbox route, I'd suggest registering a domain and then forwarding it to the url of your index page. There are free short sub-domain services out there such as .usa.cc, .co.nr etc, where you can register domains and forward them. Also, there's a free top-level domain called dot.tk where you can do the same thing, though it requires a captcha to be filled out.
Another good webhost, though probably not as good as free hosting no ads, is 0fees.net. It doesn't provide much space, but still has auto script installer with support for your basic scripts, and quick loading times.
Also, I've tried 000webhost.com, but my site was always down.
  Just some suggestions to throw out there, since I've spent 5 years with, or looking for, a website host.

Hi. I'm surprised its not working for ya...
Link is working fine here.. It just takes about 5 seconds to load.
But just in case, here's a shorter URL of the same referral link:
http://xed.cc/fhna+

I can make a "Web Archive.mht" file, if ya need me too.. thus have it open in Internet explorer, if needed.

Sadly, this is the only wayt for referring, is to use the link.

yep?
Only one there..
If it truly ain't working for ya, use the normal site link:

http://freehostingnoads.net
, and go at it the normal way.

Sorry you are having problems..

To be honest, I truly am not sure.
Back in the day, free web hosting was two GB only, and now its up to 20 GB.
Also, a VIP supporter, if ya hav 50 out of 50 referrals for it, was 10 GB, but again, its now at 20 GB.
With that being said, here's the plans that you have, besides the "free one:
http://freehostingnoads.net/webhosting.php

For a detailed explanation about this process, refer to the uploading servers topic. There is a pretty good amount of information about how this works.

http://freewha.com
unlimited bandwidth, 2 gb disc space, 12 mb filesize limit.
good hosting!

Hi,

If you do not mind, I could give you a little space on my server for you to upload files via FTP, plus a free yourname.tk domain. I could get it all set up if you like, just drop me a PM or skype, and we can set something up.