Are there free DOI generation services? Crossref, e.g., charges thousands of dollars to assign DOIs, but are there any free DOI registration services? In other words: Is there a free service that will generate DOIs for me?
As the official DOI FAQ mentions, you need to go through a recognized registration agency to get a DOI:


  
  How do I get a Digital Object Identifier (DOI) for my material?
  
  
  You must use a service offered by a DOI Registration Agency (RA). RAs
  collect metadata, assign DOI names, and offer other services such as
  reference linking or metadata lookup. See the list of RAs, and contact
  the ones whose services best meet your needs. If you do not see an
  appropriate application listed, consider approaching an existing RA or
  developing a community to build the service you require (see the DOI
  Handbook, 8 Registration Agencies, for more information). You do not
  need to be a member of the International DOI Foundation in order to
  work with an RA.


Now, you don't necessarily need to go directly through one of the listed  Registration Agencies (of which Crossref is one). Often the Registration Agencies have arrangements with other agencies to assign DOIs. (e.g. arrangements Crossref makes with publishers) If you go through those sub-agencies you can also get DOIs, though they may come with limitations.

Depending on your affiliations, there may already be a sub-agency arrangement in place. For example, the Harvard, CERN, and the British Library have arrangements in place with DataCite for providing DOIs to their materials, and there appear to be facilities in place by their national research councils for allowing DOI assigments to datasets from a number of countries.

There are a couple of places which have an "open" policy on submitting and assigning DOIs to material, regardless of affiliation. The ones I know of are Zenodo, Figshare and Dataverse. -- These are mostly based around datasets and Supporting-Material-type information (though not necessarily attached to any publication), so if you attempt to use them for other purposes you may fall afoul of their Terms of Service. They're also limited to assigning DOIs for material that is uploaded to their servers (so no DOIs for third-party websites). 

If you're looking to attach a DOI to a biology-related publication, bioArXiv assigns DOIs to submitted pre-prints (though the math/physics arXiv does not, and there are some reasons for that. - Though keep in mind that pre-publication release of articles (at least in the Biology field) may limit where you can publish.

If those options don't work for you (say you're starting a new journal and wish to assign DOIs to publications hosted on your own website), then you probably don't have much choice other than to go through one of the DOI Registration Agencies. Note that there's several of them, not just CrossRef, so it may pay to talk to others to see if you can come to an arrangement that's more suitable.