Give me more than all that you have, you can do it! Say it with me!

Question

Perhaps you've heard of me, I am the first,
Closest to winning, I am not the worst,

You think I'm the brass, nearly top of my class.
Chasing the ones before, a prize I long for.

Conducive of rounds, I will keep you up,
Beastly I'm bound, if three of me form up.

Indicating completion, I exist,
Circular and double too, I'm two twists,

Relationally similar to another here,
Find out what these lines mean, and you'll have nothing to fear.

Answer 1

The lines clue

the positive integers in increasing order.

Perhaps you've heard of me, I am the first,

$1$ is the first positive integer.

Closest to winning, I am not the worst,

$2$nd is closest to first, and according to the phrase "first the worst, second the best," second is not the worst.

You think I'm the brass, nearly top of my class.

$3$rd place receives a bronze medal; bronze and brass are both copper alloys.

Chasing the ones before, a prize I long for.

$4$th place is often the first to receive no prize.

Conducive of rounds, I will keep you up,

$5$ is the first digit that is rounded up.

Beastly I'm bound, if three of me form up.

$6$ repeated three times is the Number of the Beast in the Bible.

Indicating completion, I exist,

$7$ indicates completion in the Bible, such as in the seven days of the week.

Circular and double too, I'm two twists,

$8$ consists of two circles or twists.

Relationally similar to another here,

$9$ is a rotated version of $6$.

Find out what these lines mean, and you'll have nothing to fear.

Taking the appropriate letter of each line yields the Latin motto plus ultra meaning "further beyond," explaining the significance of the title.

Answer 2

[EDIT: Partial answer only, per OP's comment]

These are

the numbers 1 through 9

Perhaps you've heard of me, I am the first,

#1: winner

Closest to winning, I am not the worst,

#2: second place

You think I'm the brass, nearly top of my class.

#3: Bronze medal

Chasing the ones before, a prize I long for.

#4: not in the top 3

Conducive of rounds, I will keep you up,

#5 The first number that you round "up". (thanx to CGreen!)

Beastly I'm bound, if three of me form up.

#6 The number of the beast is 666 (thanx to humfuzzj)

Indicating completion, I exist,

#7 The seventh seal [biblical number of completion, per humfuzz]

Circular and double too, I'm two twists,

#8 is shaped as two circles

Relationally similar to another here,

#9 is similar-looking to "6"

Answer 3

These can be lines describing

5 vowels( a, e, i, o, u)

As,

Perhaps you've heard of me, I am the first,
Closest to winning, I am not the worst,

This is A.

You think I'm the brass, nearly top of my class. Chasing the ones before, a prize I long for.

This is for E - the most frequently used vowel among the 5 vowels and second letter (after S) in the alphabets.

Conducive of rounds, I will keep you up,
Beastly I'm bound, if three of me form up.

This is I - kind of ego, that keeps one up.

Indicating completion, I exist, Circular and double too, I'm two twists,

This is O, indicates nothingness or completion, though not sure of - twice part!

Relationally similar to another here, Find out what these lines mean, and you'll have nothing to fear.

This is for U( for YOU), and it may relate to 3rd one I, in a way.