1. AR 600-8-19: a SGT shouldn't have to ask questions about how PV2s-SFCs get promoted, a SGT should be able to answer those questions.
2. AR 623-3: every SGT gets an NCOER and will eventually write NCOERs, know what right looks like, and do your support forms on time. There is no excuse for an NCOER to be wrong, late or returned for corrections.
3. DA PAM 600-25: NCO Professional Development Guide, develop yourself and your Soldiers
4. ATP 6-22.1 – The Counseling Process: counsel your Soldiers up front about expectations, then counsel them about their development.
5. AR 600–8–2 Suspension of Favorable Personnel Actions: know the process before one of your Soldiers gets a flag and what happens next.

Extra credit
6. AR 600-20: understand Army Command Policy now.
7. AR 600-8-22: there is no rank minimum to recommend an award, if your Soldier deserves an award, you should know how to write, submit and track it now.

I'm an officer so naturally, take my recommendations with a grain of salt. Five is a little ambitious. I would focus on three and have the other two on standby to supplement what you're reading and, for when the time comes for you to actually need it, you can brush up on what you actually need and not have to read the whole thing. When you make SSG, you'll bump up you're reading. Right now, you just need the fundamentals and you need to stay focued on your people. You should also give time to counselings, taking advantage of monthly and quarterly sessions. So, don't burden yourself too much with reading regs. You'll get more into the books later, that time will come. Just a refresher before I give my list, an AR (Army regulation) tells you the rules. A DA PAM tells you how to comply with those rules and lays out a bit more praticality. An ADP (Army Doctrine Publication) addresses the fundamental ideas and principles of something. And lastly, an ATP (Army Training Publication) is focused, listing techniques and methods for the topic at hand.

1) AR 600-20: Army Command Policy
2) ADP 6-22: The Army Ethic
3) ATP 3 -21.8: Infantry Platoon and Squad
* Ranger Handbook
** DA PAM 670-1

I will also tell you not to worry about reading material on logistics right now, let your LTs worry about that. Focus on TLPs and the on-ground action of your role.

1) This is the foundation of leadership in the Army. Sit down and actually read this one, having general awareness of this will make your life easier and help you serve both your soldiers and your leaders more effectively. It will also enlighten you to the mindset of some decisions from your leadership, so you can better understand what's going on or why something is happening.
2) As the backbone of the Army, you need strong character if you're going to maintain effectiveness and add to the legacy of our institution. Give this a read and find the parts you agree and disagree with. You don't need to be a walking Uncle Sam poster, rather, you need to understand the bigger unifying vision of the Army so you can understand how you fit into the big machnice of cogs and how you will lead others into that machine.
3) We fight wars. Read this. Supplement with the Ranger handbook for proven tactics. Be the NCO that refuses to embrace POG lifestyle and never forgets the Army's mission, to fight and win our wars.
* Self-explanatory
** As I have learned from personal experience and from old seasoned NCOs, if you look sharp people are less likely to bother you. Make sure you're squared away at all times. More importantly, make sure your Joes are squared away. Given enough time, it becomes like second nature. Also, knowledge of this reg is a defensive posture. Too many people pay far too much attention to uniform standards and try and gig you or your Joes for silly stuff. Have the basics at hand so you can push through the nonsense with ease.

Lastly, pick up at least one book from someone you admire. I recommend GEN (ret.) Mattis' Call Sign Chaos or COL (ret.) David Hackworth's About Face. These are some great books and easy reads, great for when you're sitting around somewhere. They don't take much brain power, they're entertaining, and they teach you stuff. You're going to get tested as you and try and make E5. People will ask you to explain why you did something or why you made such and such decision. Referring to historical event as an example, or to the opinion of some senior ranking figure that you admire, is an admirable and professional way of developing yourself and showing others that you take yourself and your work seriously.

https://www.amazon.com/About-Face-Odyssey-American-Warrior/dp/ [login to see]
https://www.amazon.com/Call-Sign-Chaos-Learning-Lead/dp/ [login to see]

Unfortunately, you need more than 5. The good news is, you can always reference. Knowing WHERE the information can be found is, in most cases, as good as knowing the information. The difference us going to be in on-the-spot things. For that reason I go with the following:

AR 670-1. I cannot count the number of times I have heard an NCO say they wanted to make a correction, but were not quite sure what the right thing was. Know both the male AND female requirements - and stay up to date with MILPERs and ALARACTs that may issue updates.

AR 600-20. Know what authority you have, how you get it, and who has the level of authority you need for problems above your pay grade.

AR 25-20. Like it or not, at least 1/3 of your job now consists of documenting and corresponding. Do it write (pun intended).

AR 385-10. Safety is everyone's job. Know the rules.

Barracks SOP. Most of the time this is a garrison-level policy. But find it and know it. Not barracks lawyer know it, KNOW IT know it. At some point in time, you WILL have a dispute over barracks rules. Know the answer ahead of time.

Others have also given you some great suggestions. As I mentioned, there are more than 5. I recommend starting with these 5 because they are (IMHO) the most likely to be needed in the moment, without having time to go back and reference.