To a precision of 1 decimal digits, the maximum number you can work with is 562949953421311.

To a precision of 2 decimal digits, it's 70368744177663. Interestingly, the first number is equal to:

(Number.MAX_SAFE_INTEGER + 1) / 16 - 1

And the second number is equal to:

(Number.MAX_SAFE_INTEGER + 1) / 128 - 1

What we're looking for, is the maximum safe number to support a precision of d digits after the decimal point. By "support" I mean "can reliably do basic arithmetic".

For example, we know that Number.MAX_SAFE_INTEGER (aka 2**53-1) is not safe, because basic arithmetic is broken:

Number.MAX_SAFE_INTEGER - 0.1 === Number.MAX_SAFE_INTEGER
>>> true // unsafe

And we know that 0 is safe, since:

0 + 0.1 === 0
>>> false // safe

BTW, 0 is reliable as far as 1e-323 (including):

0 + 1e-323 === 0
>>> false // safe

0 + 1e-324 === 0
>>> true // unsafe

I binary-searched between 0 and Number.MAX_SAFE_INTEGER for the biggest number that answers that definition, and came up with these numbers.

Here's the code (pass any other number to findMaxSafeFloat() at the end of snippet)

/**Returns whether basic arithmetic breaks between n and n+1, to a precision of `digits` after the decimal point*/
function isUnsafe(n, digits) {
  // digits = 1 loops 10 times with 0.1 increases.
  // digits = 2 means 100 steps of 0.01, and so on.
  let prev = n;
  for (let i = 10 ** -digits; i < 1; i += 10 ** -digits) {
    if (n + i === prev) { // eg 10.2 === 10.1
      return true;
    }
    prev = n + i;
  }
  return false;


}

/**Binary search between 0 and Number.MAX_SAFE_INTEGER (2**53 - 1) for the biggest number that is safe to the `digits` level of precision.
 * digits=9 took ~30s, I wouldn't pass anything bigger.*/
function findMaxSafeFloat(digits, log = false) {
  let n = Number.MAX_SAFE_INTEGER;
  let lastSafe = 0;
  let lastUnsafe = undefined;
  while (true) {
    if (log) {
      console.table({
        '': {
          n,
          'Relative to Number.MAX_SAFE_INTEGER': `(MAX + 1) / ${(Number.MAX_SAFE_INTEGER + 1) / (n + 1)} - 1`,
          lastSafe,
          lastUnsafe,
          'lastUnsafe - lastSafe': lastUnsafe - lastSafe
        }
      });
    }
    if (isUnsafe(n, digits)) {
      lastUnsafe = n;
    } else { // safe
      if (lastSafe + 1 === n) { // Closed in as far as possible
        console.log(`\n\nMax safe number to a precision of ${digits} digits after the decimal point: ${n}\t((MAX + 1) / ${(Number.MAX_SAFE_INTEGER + 1) / (n + 1)} - 1)\n\n`);
        return n;
      } else {
        lastSafe = n;
      }
    }
    n = Math.round((lastSafe + lastUnsafe) / 2);
  }
}

console.log(findMaxSafeFloat(1));

An interesting thing I've found by lining up the safe numbers, is that the exponents don't step up in a consistent manner. Look at the table below; once in a while, the exponent increases (or decreases) by 4, and not 3. Not sure why.

| Precision | First UNsafe                | 2^53/x                   |
|-----------|-----------------------------|--------------------------|
| 1         | 562,949,953,421,312 = 2^49  | x = 16 = 2^4             |
| 2         |  70,368,744,177,664 = 2^46  | x = 128 = 2^7            |
| 3         |   8,796,093,022,208 = 2^43  | x = 1,024 = 2^10         |
| 4         |     549,755,813,888 = 2^39  | x = 16,384 = 2^14        |
| 5         |      68,719,476,736 = 2^36  | x = 131,072 = 2^17       |
| 6         |       8,589,934,592 = 2^33  | x = 1,048,576 = 2^20     |
| 7         |         536,870,912 = 2^29  | x = 16,777,216 = 2^24    |
| 8         |          67,108,864 = 2^26  | x = 134,217,728 = 2^27   |
| 9         |           8,388,608 = 2^23  | x = 1,073,741,824 = 2^30 |

It's pretty much the same in JavaScript, only that min and max are members of the Math object:

var n = Math.min(Math.max(n, 1), 20);

JavaScript has two number types: Number and BigInt.

The most frequently-used number type, Number, is a 64-bit floating point IEEE 754 number.

The largest exact integral value of this type is Number.MAX_SAFE_INTEGER, which is:

• 2⁵³-1, or
• +/- 9,007,199,254,740,991, or
• nine quadrillion seven trillion one hundred ninety-nine billion two hundred fifty-four million seven hundred forty thousand nine hundred ninety-one

To put this in perspective: one quadrillion bytes is a petabyte (or one thousand terabytes).

"Safe" in this context refers to the ability to represent integers exactly and to correctly compare them.

From the spec:

Note that all the positive and negative integers whose magnitude is no greater than 2⁵³ are representable in the Number type (indeed, the integer 0 has two representations, +0 and -0).

To safely use integers larger than this, you need to use BigInt, which has no upper bound.

Note that the bitwise operators and shift operators operate on 32-bit integers, so in that case, the max safe integer is 2³¹-1, or 2,147,483,647.

const log = console.log
var x = 9007199254740992
var y = -x
log(x == x + 1) // true !
log(y == y - 1) // also true !

// Arithmetic operators work, but bitwise/shifts only operate on int32:
log(x / 2)      // 4503599627370496
log(x >> 1)     // 0
log(x | 1)      // 1

Technical note on the subject of the number 9,007,199,254,740,992: There is an exact IEEE-754 representation of this value, and you can assign and read this value from a variable, so for very carefully chosen applications in the domain of integers less than or equal to this value, you could treat this as a maximum value.

In the general case, you must treat this IEEE-754 value as inexact, because it is ambiguous whether it is encoding the logical value 9,007,199,254,740,992 or 9,007,199,254,740,993.