I've seen where companies have filed for the same patent in multiple additional countries beyond the United States. What is the motivation for such extensive patenting? What impact (if any) do these international patents have on their US patent? What strategy is used to decide what other countries are worth patenting in and what other countries are just skipped? Are their reciprocal agreements with some countries that make such additional patents needless or even with reciprocality is it sometimes advantageous to still apply for the other countries patent as well?
Unlike copyright, a patent is only valid in a country where it has been filed, accepted and registered. A US patent gives you an exclusive right to use your invention (including manufacturing a product using it and selling a product manufactured using it) in the US. It doesn't constrain someone who operates entirely outside the US.
There are some groups of countries that have mutual patent recognition agreements (ARIPO and OAPI in Africa, EAPO around Russia, EPO in Europe). These groups of countries have agreements where a patent application is examined by only one office, but if the patent is granted the applicant must decide in which countries in the group the patent will apply (and pay each country's fee). The US isn't a member of any such organization; there's no particular agreement on patents between the US and Canada, for example.
In order to apply in more than one country (or group of countries), the patent must be examined in each country. A majority of countries (including the US) have ratified the Patent Cooperation Treaty. The usual way to extend a patent to many countries is to file it in your home country, then file a PCT application, then have the patent examined by national offices of PCT member states. See the PCT website for more information. See also this answer. Note that it can happen that a patent is granted in one country but rejected in another: maybe one country will find some prior art that another country won't, or maybe they'll interpret it in a different way.
Which countries to file your patent is depends heavily on what the logistics and market is for your product. If you only ever expect to sell your product in the US, a US patent is good enough. If you think your product can only be manufactured economically in a few countries, patent it in these countries (assuming that you trust these countries' patent enforcement). If you think your product could be manufactured and sold anywhere, it's a matter of investment: each additional country costs you money upfront but locks your competitors down more.