Q:

Is there a way to count with macros?

I want to create a macro that prints "Hello" a specified number of times. It's used like:
many_greetings!(3);  // expands to three `println!("Hello");` statements

The naive way to create that macro is:
macro_rules! many_greetings {
    ($times:expr) => {{
        println!("Hello");
        many_greetings!($times - 1);
    }};
    (0) => ();
}

However, this doesn't work because the compiler does not evaluate expressions; $times - 1 isn't calculated, but fed as a new expression into the macro.

A:

While the ordinary macro system does not enable you to repeat the macro expansion many times, there is no problem with using a for loop in the macro:
macro_rules! many_greetings {
    ($times:expr) => {{
        for _ in 0..$times {
            println!("Hello");
        }
    }};
}

If you really need to repeat the macro, you have to look into procedural macros/compiler plugins (which as of 1.4 are unstable, and a bit harder to write).
Edit: There are probably better ways of implementing this, but I've spent long enough on this for today, so here goes. repeat!, a macro that actually duplicates a block of code a number of times:
main.rs
#![feature(plugin)]
#![plugin(repeat)]

fn main() {
    let mut n = 0;
    repeat!{ 4 {
        println!("hello {}", n);
        n += 1;
    }};
}

lib.rs
#![feature(plugin_registrar, rustc_private)]

extern crate syntax;
extern crate rustc;

use syntax::codemap::Span;
use syntax::ast::TokenTree;
use syntax::ext::base::{ExtCtxt, MacResult, MacEager, DummyResult};
use rustc::plugin::Registry;
use syntax::util::small_vector::SmallVector;
use syntax::ast::Lit_;
use std::error::Error;

fn expand_repeat(cx: &mut ExtCtxt, sp: Span, tts: &[TokenTree]) -> Box<MacResult + 'static> {
    let mut parser = cx.new_parser_from_tts(tts);
    let times = match parser.parse_lit() {
        Ok(lit) => match lit.node {
            Lit_::LitInt(n, _) => n,
            _ => {
                cx.span_err(lit.span, "Expected literal integer");
                return DummyResult::any(sp);
            }
        },
        Err(e) => {
            cx.span_err(sp, e.description());
            return DummyResult::any(sp);
        }
    };
    let res = parser.parse_block();

    match res {
        Ok(block) => {
            let mut stmts = SmallVector::many(block.stmts.clone());
            for _ in 1..times {
                let rep_stmts = SmallVector::many(block.stmts.clone());
                stmts.push_all(rep_stmts);
            }
            MacEager::stmts(stmts)
        }
        Err(e) => {
            cx.span_err(sp, e.description());
            DummyResult::any(sp)
        }
    }
}

#[plugin_registrar]
pub fn plugin_registrar(reg: &mut Registry) {
    reg.register_macro("repeat", expand_repeat);
}

added to Cargo.toml
[lib]
name = "repeat"
plugin = true

Note that if we really don't want to do looping, but expanding at compile-time, we have to do things like requiring literal numbers. After all, we are not able to evaluate variables and function calls that reference other parts of the program at compile time.

A:

As far as I know, no. The macro language is based on pattern matching and variable substitution, and only evaluates macros.
Now, you can implement counting with evaluation: it just is boring... see the playpen
macro_rules! many_greetings {
    (3) => {{
        println!("Hello");
        many_greetings!(2);
    }};
    (2) => {{
        println!("Hello");
        many_greetings!(1);
    }};
    (1) => {{
        println!("Hello");
        many_greetings!(0);
    }};
    (0) => ();
}

Based on this, I am pretty sure one could invent a set of macro to "count" and invoke various operations at each step (with the count).

A:

As the other answers already said: no, you can't count like this with declarative macros (macro_rules!). 

But you can implement the many_greetings! example as a procedural macro. procedural macros were stabilized a while ago, so the definition works on stable. However, we can't yet expand macros into statements on stable -- that's what the #![feature(proc_macro_hygiene)] is for.
This looks like a lot of code, but most code is just error handling, so it's not that complicated!
examples/main.rs
#![feature(proc_macro_hygiene)]

use count_proc_macro::many_greetings;

fn main() {
    many_greetings!(3);
}

Cargo.toml
[package]
name = "count-proc-macro"
version = "0.1.0"
authors = ["me"]
edition = "2018"

[lib]
proc-macro = true

[dependencies]
quote = "0.6"

src/lib.rs
extern crate proc_macro;

use std::iter;
use proc_macro::{Span, TokenStream, TokenTree};
use quote::{quote, quote_spanned};

/// Expands into multiple `println!("Hello");` statements. E.g.
/// `many_greetings!(3);` will expand into three `println`s.
#[proc_macro]
pub fn many_greetings(input: TokenStream) -> TokenStream {
    let tokens = input.into_iter().collect::<Vec<_>>();

    // Make sure at least one token is provided.
    if tokens.is_empty() {
        return err(Span::call_site(), "expected integer, found no input");
    }

    // Make sure we don't have too many tokens.
    if tokens.len() > 1 {
        return err(tokens[1].span(), "unexpected second token");
    }

    // Get the number from our token.
    let count = match &tokens[0] {
        TokenTree::Literal(lit) => {
            // Unfortunately, `Literal` doesn't have nice methods right now, so
            // the easiest way for us to get an integer out of it is to convert
            // it into string and parse it again.
            if let Ok(count) = lit.to_string().parse::<usize>() {
                count
            } else {
                let msg = format!("expected unsigned integer, found `{}`", lit);
                return err(lit.span(), msg);
            }
        }
        other => {
            let msg = format!("expected integer literal, found `{}`", other);
            return err(other.span(), msg);
        }
    };

    // Return multiple `println` statements.
    iter::repeat(quote! { println!("Hello"); })
        .map(TokenStream::from)
        .take(count)
        .collect()
}

/// Report an error with the given `span` and message.
fn err(span: Span, msg: impl Into<String>) -> TokenStream {
    let msg = msg.into();
    quote_spanned!(span.into()=> {
        compile_error!(#msg);
    }).into()
}

Running cargo run --example main prints three "Hello"s. 

