I have not used Dreamhost or Dream"Press", but if you are looking for solid service, go with an actual WordPress host, not a host that does WordPress on the side (host/wp or host/wordpress). A host offering WordPress on the side is basically giving you the same service as their regular server.
A few things to also take into account;
1. Update everything. Sounds easy but some will go forever and never update anything
2. Have security plugins in place. WordFence is a good start
3. Most WP specific hosts use Nginx. Serves pages quickly and uses less resources than the standard Apache/centos/cpanel
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If you test DreamPress, please report back to WHT. I'm interested, but it's very unlikely that I'd try this myself.
My choices would be a SSD-powered host (EuroVPS is close enough to my location), or just any generic cPanel host with good reputation.
Dreamhost is very good with scripting and automation. I'm sure they can get hardening done - they've got the resources. Their VPS offering can change to nginx with one click. If they're running that service with much lighter loads than their usual front-end boxes, I see the often-saturated network as the biggest potential problem. So the basic building blocks are there, but it's impossible to say if a budget provider can run a solid WordPress hosting service.
I decided to go with WP Engine. WP Engine seems very good so far. Just a bit pricey. A bit more pricey than I think I can handle with their 1,000 visitors for $1 overage fee if I go beyond my 25,000 visitors (which includes bot visits and all)