Why can't I inherit from int in C++?

Neil's comment is pretty accurate. Bjarne mentioned considering and rejecting this exact possibility¹:

The initializer syntax used to be illegal for built-in types. To allow it, I introduced the notion that built-in types have constructors and destructors. For example:

int a(1);    // pre-2.1 error, now initializes a to 1

I considered extending this notion to allow derivation from built-in classes and explicit declaration of built-in operators for built-in types. However, I restrained myself.

Allowing derivation from an int doesn't actually give a C++ programmer anything significantly new compared to having an int member. This is primarily because int doesn't have any virtual functions for the derived class to override. More seriously though, the C conversion rules are so chaotic that pretending that int, short, etc., are well-behaved ordinary classes is not going to work. They are either C compatible, or they obey the relatively well-behaved C++ rules for classes, but not both.

As far as the comment the performance justifies not making int a class, it's (at least mostly) false. In Smalltalk all types are classes -- but nearly all implementations of Smalltalk have optimizations so the implementation can be essentially identical to how you'd make a non-class type work. For example, the smallInteger class is represents a 15-bit integer, and the '+' message is hard-coded into the virtual machine, so even though you can derive from smallInteger, it still gives performance similar to a built-in type (though Smalltalk is enough different from C++ that direct performance comparisons are difficult and unlikely to mean much).

The one bit that's "wasted" in the Smalltalk implementation of smallInteger (the reason it only represents 15 bits instead of 16) probably wouldn't be needed in C or C++. Smalltalk is a bit like Java -- when you "define an object" you're really just defining a pointer to an object, and you have to dynamically allocate an object for it to point at. What you manipulate, pass to a function as a parameter, etc., is always just the pointer, not the object itself.

That's not how smallInteger is implemented though -- in its case, they put the integer value directly into what would normally be the pointer. To distinguish between a smallInteger and a pointer, they force all objects to be allocated at even byte boundaries, so the LSB is always clear. A smallInteger always has the LSB set.

Most of this is necessary, however, because Smalltalk is dynamically typed -- it has to be able to deduce the type by looking at the value itself, and smallInteger is basically using that LSB as a type-tag. Given that C++ is statically typed, there's never a need to deduce the type from the value, so you probably wouldn't need to "waste" that bit on a type-tag.

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^{1. In The Design and Evolution of C++, §15.11.3.}

Int is an ordinal type, not a class. Why would you want to?

If you need to add functionality to "int", consider building an aggregate class which has an integer field, and methods that expose whatever additional capabilities that you require.

Update

@OP "Ints aren't classes" so?

Inheritance, polymorphism and encapsulation are keystones of object oriented design. None of these things apply to ordinal types. You can't inherit from an int because it's just a bunch of bytes and has no code.

Ints, chars, and other ordinal types do not have method tables, so there's no way to add methods or override them, which is really the heart of inheritance.

Why would I want to? Stronger typing. For example, I could define two classes intA and intB, which let me do intA+intA or intB+intB, but not intA+intB.

That makes no sense. You can do all that without inheriting from anything. (And on the other hand, I don't see how you could possibly achieve it using inheritance.) For example,

class SpecialInt {
 ...
};
SpecialInt operator+ (const SpecialInt& lhs, const SpecialInt& rhs) {
  ...
}

Fill in the blanks, and you have a type that solves your problem. You can do SpecialInt + SpecialInt or int + int, but SpecialInt + int won't compile, exactly as you wanted.

On the other hand, if we pretended that inheriting from int was legal, and our SpecialInt derived from int, then SpecialInt + int would compile. Inheriting would cause the exact problem you want to avoid. Not inheriting avoids the problem easily.

"Ints don't have any member functions." Well, they have a whole bunch of operators like + and -.

Those aren't member functions though.