Short answer:

Here are the indisputable facts. You can decide yourself how many times you think the votes have been counted and what constitutes a count under Florida law.

1) Under Florida law Section 101.5614 (titled Canvassing of Returns), the counting of votes includes running the ballots through the counting machines and a manual inspections of any ballots not "properly counted by machines" to determine if a voter intent can clearly be determined.  In other words, a full count under Florida law requires both a machine count and manual counts of any ballots not read by machine on which a clear indication of voter intent can be determined by the canvassing boards.
2) On election night, every county ran their ballots through the machines, but they did not conduct manual inspections of ballots not read by the machines.
3) Pursuant to Florida law, all the votes were supposed to be recounted. However, only 10 counties actually ran all their ballots back through the machines and manually inspected machine unreadable ballots, both of which are required counting procedures under Florida law.  The other 57 counties did something less, including 16 that did absolutely no recounting whatsoever.

4)  Two counties, Broward and Volusia, finished full manual recounts of every ballot pursuant to protest and manual recount provisions under 102.166 and had those recounts included in their returns.

Yes, all the votes were counted by machine once, and then about half to two-thirds were counted by machines a second time.
However, Florida law does not consider only a machine count of the ballots a "count."  Machine counting is only one part of the counting process under Florida Law.   The other part is manual inspections of machine unreadable ballots.  Only 12 counties representing roughly a third of all Florida's ballots were put through what Florida law considers a count.
 

Long answer (this was written prior to the Florida Supreme Courts order to count all the undervotes statewide):

Saying hand counts across Florida are unnecessary because the votes have been counted and recounted is patently false.
Many counties did not actually recount their ballots, settling instead to simply report their original count (not surprisingly many  of those were Republican counties).  More importantly, tens of thousands of ballots across the state have yet to be counted,   either the during the first count or the limited recount.  Why?  It is not because they don't have a vote on them, but because the machines can't read them.

Of course, this happens in every election -- machines fail to read ballots that contain votes.  It is not a travesty of Democracy  to leave those ballots uncounted in most cases because the margin of victory is large enough that the "will of the people" is  obvious.  But when the margin of victory is well within the margin of error caused by machine errors, ballot errors, and voter  errors, the next step must be to try and count the yet uncounted votes so as many people as possible have an opportunity to   participate in our representative Democracy.

Having the right to vote means nothing unless your vote is actually counted.  Any government can hold a vote, but only a  Democracy dictates that those votes are actually counted.  Now, we have a situation in which tens of thousands of people,  Republicans and Democrats alike, stood in line and exercised their right to vote.  They cast votes under the assumption that   their vote counts.  Yet, because the ballots they voted on are less than perfect and the machines used to count those votes are  less than perfect, their votes are being ignored.  When races are close, the universally preferable method (even in Texas) for determining the votes on machine unreadable ballots is a hand count.

It is true that Bush lead after the first vote by 1,700 votes, and is still in the lead by about 900 after the second count of about  half the ballots were added to the overseas ballots.  Should the remaining uncounted ballots remain as such simply because the Bush Campaign and the Florida Secretary of State are pleased with the current count? Should we only try to count those  uncounted votes in largely Democratic counties so Gore can find enough votes to win?  If we let Democracy dictate our   actions rather than partisan politics, the next step must be to count, for the first time, those tens of thousand of ballots across  the state on which the people have voiced their preference only to have it silenced by machine error.  Of course, Gore eventually got with the program and came out in favor of this approach, but only because he knows that Bush has pinned   himself into a strategy that prevents him from advocating any hand counts (except I guess the one occurring right now in   Northeast Texas).  However, Gore showed that his position on this point was purely rhetorical, because he never filed a single lawsuit to try and get all these votes counted.  While there was nothing he could do about during the protest phase of the election, he could certainly have argued to have all the votes counted under the contest phase.  Instead, he chose only to try and get Dade's uncounted votes examined.

The message from Bush: let the Secretary of State use her "discretionary power" to stop the democratic process of vote  counting and declare me winner because I am in the lead, even though tens of thousands of votes have never been counted.  The message from Gore: let the State Supreme Court allow a count of those uncounted votes in counties were I did well so I   can overtake Bush, even though thousands of votes will remain uncounted.  Personally, I prefer to let Democracy run its   course by counting as many uncounted votes as possible and let those tells us who should win Florida rather than the Secretary of State, or the State Supreme Court, or the Bush Campaign, or three counties in South Florida, or the Gore campaign.  I fear that such a fair and accurate solution is now beyond the realm of possibility.