You can take unlimited trips during the year, www.carinsurent.com insurance policies allow you to travel for up to 45 days. For shorter rentals you can buy daily cover for US$ 5.50.
I have contacted the company listed by chang to verify that the company DOES in fact cover non-EU citizens renting cars in Ireland and the United Kingdom. I will advise after receiving their reply.
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Bob
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Heard back today by email from the company that they DO offer 'Excess' coverage to non-EU citizens -- and, in particular, those from North America. NOT an endorsement, per se, as I've NO personal experience, but I think it's a good sign.
Back in 2009, there were half a dozen companies, but by 2016, I only knew of one: https://www.worldwideinsure.com/annual-travel-insurance.htm