I heard this phrase a few times but I still can't grasp its meaning. Does it mean, "as much as you would like"?
You can translate it as "Knock yourself out". It means "Go ahead and do that thing, if you want to". Sometimes it's said sarcastically, as if the thing the other person wants to do/try isn't going to work out. Sometimes it's just used as "Please, go ahead", without a sarcastic tone. If there is an exclamation mark after it (or the spoken equivalent) it's probably sarcastic.
I usually see it written as 思い切ってどうぞ.
思い切る, as you might guess from the letters, literally means "to stop deliberating/thinking". It can actually have two meanings: "to give up", or to "decide". That said, when it's used as 「思いっきり」, it always means to do something decisively, with gusto and without reservation.
思いっきり has been idiomatized to the point that it no longer necessarily modifies a verb; it can serve to emphasize an adjective as well: 「彼は思いっきり弾けた人格だ」(he's got a extremely spontaneous personality).