Variables are available only in the scope you defined them. If you define a variable inside a function, you won't be able to access it outside of it.
Define variable with var outside the function (and of course before it) and then assign 10 to it inside function:
var value;
$(function() {
value = "10";
});
console.log(value); // 10
Note that you shouldn't omit the first line in this code (var value;), because otherwise you are assigning value to undefined variable. This is bad coding practice and will not work in strict mode. Defining a variable (var variable;) and assigning value to a variable (variable = value;) are two different things. You can't assign value to variable that you haven't defined.
It might be irrelevant here, but $(function() {}) is a shortcut for $(document).ready(function() {}), which executes a function as soon as document is loaded. If you want to execute something immediately, you don't need it, otherwise beware that if you run it before DOM has loaded, value will be undefined until it has loaded, so console.log(value); placed right after $(function() {}) will return undefined. In other words, it would execute in following order:
var value;
console.log(value);
value = "10";
See also: