You can do nearly anything, but it's ultimately up to the DM.
With a minimum of 15 from glibness, +6 from Charisma, +6 from proficiency and another +6 from expertise in that proficiency, you're rolling a minimum of 33 and a maximum of 38. Prodigy only gives expertise in one skill, so you'd need a second source of expertise. You also have to take a round to cast glibness, which lasts only an hour and can be cast only once per long rest.
However, this is entirely do-able, and your skill would be effective against all but the most extreme challenges.
The difficulty class of ability checks are ultimately determined by the DM. The Dungeon Master's Guide, p.238 says only that such a check is nearly impossible:
"A DC 30 check is nearly impossible for most low-level characters. A 20th-level character with proficiency and a relevant ability score of 20 still needs a 19 or 20 on the die roll to succeed at a task of this difficulty."
The primary use of a reliable 30+ minimum check result is that you can reliably pass "nearly impossible" tasks, which are the hardest type of check defined in the rules. The only checks you can't pass on a roll of 33 are things which are either literally impossible (in which case no result would ever succeed), checks which the DM has decided are even harder than nearly impossible (no rules define what these are), and opposed rolls against similarly skilled opponents (which I'll cover later).
There are no rules in 5e for exceptional successes by exceeding the task by a high margin (though some DMs will allow this).
Page 245 gives the rules for Charisma checks to adjust a creature's attitude with conversation, which may allow the use of Persuasion or Deception if the DM considers it relevant. A DC 20 check, which you will automatically pass, allows you to:
- Convince a hostile creature to do as you wish, but only if there is no risk or cost to itself
- Convince an indifferent creature to do as you wish, even if it means accepting minor risk
- Convince a friendly creature to do as you wish, even at significant risk or cost to itself
Personally, in past campaigns, I've allowed 30+ Persuasion checks to achieve incredible things beyond this, such as convincing a hostile creature to become an ally, or an enemy to betray their superior and switch sides. The only rule supporting this is that things which are nearly impossible are DC 30.
However, as per DMG p. 244, "Starting Attitude", it's possible for a creature to be so actively hostile that Charisma checks automatically fail, so rules-as-written, you cannot count on it to automatically win every fight by talking it out. "Hostile" is defined as someone who is your enemy, but not necessarily someone who is attacking you right now. I get the sense that you can't talk your way out of a fight against someone who's not willing to stop and listen.
Skills can also be used in opposed rolls (DMG p.238), in which two people make opposed checks. Persuasion might be used this way in a diplomatic negotiation or court of law, where you would frequently win because your opposed check would almost always beat anyone else's. Your character would be highly sought after as an ambassador, and highly rewarded for that.
Looking also at the Monster Manual and other sourcebooks, with a roll of 33-38, you could competently out-debate an ancient gold dragon (Persuasion +16), whose range of rolls is 17-36. Likewise an empyrian (Persuasion +15), Graz'zt, Prince of Demons (Persuasion +15), or an illithid Elder Brain (Persuasion +12). You can lie to Acererak (Insight +12) with certainty of success. Not only would kings seek your diplomatic services, gods would too.
The most powerful creature that 5e can make a statblock for would be CR 30 with an ability score of 30, having therefore a +9 proficiency bonus and a +10 modifier, for +19, a roll range of 20-39. You would frequently succeed at Persuasion or Deception against a titan-level quasi-deity. You could even fool Tiamat (a full deity, and one of the most powerful creatures in D&D), but not a Greater Deity, who in 5e are officially beyond mortal limitations and don't even have statblocks.