Right now as it stands, the A's have the 25th, 62nd, 71st, 100th, and 106th picks. Under the new Collective Bargaining Agreement (CBA), the A's #1 draft pick can potentially move up 4 spots to the #21 slot. That's because four free agents are left that have draft pick compensation tied to them: Michael Bourn, Adam Laroche, Kyle Lohse, and Rafael Soriano.
The way draft pick compensation works under the new CBA and why the A's can potentially move up four slots is that when a player is offered a qualifying offer, something around $13.5 million this offseason but is subject to change other offseasons, he can deny it and thus be tied to a draft pick. If a team signs that player, they lose/forfeit their pick, however under the new CBA, the team that made the qualifying offer doesn't gain the 1st or 2nd round pick, just a supplemental round pick which comes right at the end of the first round. Basically the pick disappears into thin air. That's what happened when the Angels signed Josh Hamilton, they forfeited their pick but unlike under the last CBA, Texas didn't gain their pick, instead they only get a supplemental round pick. The top 10 picks under the new rules are protected, before it was the top 15 picks. This is why the Indians don't forfeit their first pick. Instead, their 2nd round pick disappears and the Yankees get just the supplemental round pick (they're eligible for 3 supplemental round picks, yeesh).
So since I have too much time on my hands this offseason, just thought I'd do a little digging around and looked up the draft picks from each slot I have listed above. I'll put the highlights from each slot, if there are any. There are a lot of ways to define a "highlight" pick, but I'll just roughly define it as at least 1-rWAR (since I'm using Baseball-Reference to look up the picks anyway and I'm lazy). Obviously getting 1-rWAR from a 25th pick isn't as impressive as getting it from a 100th pick, but it still means the player made it to the bigs and even if he wasn't good, he still stayed there for some amount of time.
A's pick(s): Brad Sullivan, Bobby Crosby, John Wasdin.
So the pick hasn't exactly been friendly to the A's in terms of major league production, but it's produced some damn good talent.
Fun fact: Now John Wasdin is their low-A (Beloit Snappers) pitching coach and Todd Steverson, drafted 25th in 1992 by the Blue Jays, is their minor league hitting coordinator.
A's pick(s): Bruce Maxwell, Andre Ethier, Steven Tingle, Bill Cotton.
So we had a really good pick who turned out to be very successful in the majors. Unfortunately it wasn't for the A's. However, because the A's traded him away, they made the playoffs in 2006. Hopefully their most recent #62 pick can have sustained success in the majors. Overall the talent from that pick isn't a whole lot to write home about.
A's pick(s): Floyd Bannister
First of all: Wow...that's a lot of talent to come out of this pick. There's three 20 rWAR guys, one who'll probably be that by the end of next season, and another guy who will probably be a 10 rWAR guy by the end of his career. As for the A's, one of those 20 rWAR guys was their lone pick at this slot.
Fun fact: Yes, in fact, the Bob Melvin listed above is the same Bob Melvin that currently manages the A's.
Highlights: Josh Bard, Ron Gant, Shane Mack, George Vukovich
A's pick(s): none
So not a whole lot out of this pick, but the two who were truly successful, Gant and Mack, put up 30+ and 20+ rWAR respectively.
A's pick(s): Kyle Twomey, Brian Graham.
Obvious talent is obvious. One should-be HOF and another possible HOF. The rest, meh.
Side note: While the A's drafted Twomey last year, they didn't sign him. Their (currently) 106th slot is the compensation for not signing him.
I must reiterate that these are not the final Athletics draft slots due to the new CBA. Their picks can move up anywhere from 0 to 4 slots.