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I have used GoDaddy for clients that need a lot of hand holding. Their customer support is geared towards clients that need a lot of assistance. Takes the load off me. Not the cheapest option for them, but definitely for you.
I know is a long shot. But .netcore runs in Linux. You might want to look into Digital Ocean. I did some early development with aspnet core 1.0 and it was fast and responsive with a Debian droplet. All experimental. It was interesting.
Thanks for the suggestion, but I need stable, not experimental. A Raspberry PI was suggested up stream, and THAT is an entertaining thought ... and dirt cheap for the client. But they need something reliable so some type of service is better for everyone.
However, some time down the road when I've had a chance to experiment with my PI's ... I may choose otherwise.
Arvixe I use a good bit. Very reasonable and while it requires that you have some reasonable level of knowledge. They have been the most cost effective .NET host provider I've run into. When there has been a real problem they were capable and resolved issues in a reasonable length of time.
They have both shared and dedicated hosting at various levels based on your need and very quickly get to unlimited storage and SQL backends. MySQL is provided by default with all of their packages.
I also use SmarterAsp.net. I use webhostforasp.net as a secondary provider. I received free package deals for both and have kept them around because why get rid of free, but that was back when they were just starting out. I have had good luck with both of them. Easy control panels, easy to load sites and fairly easy to manage.
I would recommend you look at Azure as well. I wouldn't look in any other direction if you are using a .net application as you will probably have too many problems maintaining everything. That is my practical experience, anyway.
The problem I have with all of the Cloud solutions is the true hidden cost. If you purchase a hosting account through one of the providers above, you get x space, with y sites, with z databases and everything is spelled out. The last time I looked at Cloud solutions you get some of the same but some of the billing is based on usage volume, number of Mbytes written to the disk or along the lan etc. Before recommending either or choosing a direction, I would see if you could get a true hard fixed, per month cost. That may help. Your client isn't going to care about "if I use this much volume in March, it costs $100, if I use this much volume in April it will cost $200", they are just going to be pissed when their bill doubles. Plus, I have a golden rule that I always use for applications, and that is they will always grow. If you are only using a little bit now, give it a couple years and that will be significantly larger.
The problem I have with all of the Cloud solutions is the true hidden cost.
Costs are a driver, but hidden costs are a more dangerous point in general. I'm thinking of not only hosting costs, but my time to manage anything they can't (which appears likely). I expect the client will be willing to pay a bit more for stable service that doesn't cost them for my time -- which will be cheaper for them. [Plus I long ago lost interest in waking up at 2 AM to handle a production problem. ]
I have some "business card" sites that are hosted on SmarterAsp / MyWindowsHosting as the price is right. They do have a IIS Application Sleep that is a little slow to respond which I took care of by setting up a Pingdom account to hit the site every 5 minutes.
I use to use CrystalTech without issue, actually preferred them but they started bumping up their prices and getting out of shared hosting.
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