Shared hosting can only handle so many concurrent requests. Depending on how heavy the traffic is on your site, or even on all sites hosted on the same server, you will experience slowness. Not all hosts allow minute cron. You won't have as much control over how PHP/Apache is configured, you probably won't be able to install server-side caching tools (i.e. memcached) etc. It's probably fine for development purposes, but I wouldn't host anything on GoDaddy. I have been using A2Hosting for some private Elgg hosting and it does the job. I only have hateful things to say about GoDaddy (not to mention an arse of a CEO that they have)

Lol. I know what you mean. Can't tell you how many times I wanted to switch services, but now I'm comfortable with it. (Got any other suggestions? I tried Dreamhost, but don't really like it.) Anyway, digging a little further, I found that GoDaddy now has elgg support. They offer 2.3.1. 

Depends on what you want to spend. 
I would recommend going for one of those fancy cloud VPS servers and set a cheap one up for development and scale when your website grows larger.

There are plenty good ones starting from around 5$ a month.
My development and startup server is at digitalocean. I plan to scale it there depending on how the website will evolve. No intension in advertising, but I had nothing then good experience there. 

Downside is that you need 'some' knowledge about linux or shell access, but most of those providers have ready-made images for you with all the whistles and bells installed in just one click.
That downside becomes a plus when you realize that you can control about anything in how to your server is configured, as Ismayil said.

Already trying for 30 days! Thats a long time. Maybe your IT guy could share the problems he is facing on this community so maybe we could help him. It would also help to describe your setup. Is it a WAMP installation or are you using IIS or something else...