Change click() to almost anything but that. I suggest incrementClick(). click() is a reserved function in JavaScript.
<html>
<head>
    <title>Space Clicker</title>
</head>
<body>
    <script type="text/javascript">
    var clicks = 0;
    function incrementClick() {
        clicks += 1;
        document.getElementById("clicks").innerHTML = clicks;
    };
    </script>
    <button type="button" onClick="incrementClick()">Click me</button>
    <p>Clicks: <a id="clicks">0</a></p>
</body></html>
For persisting the clicks across page refresh, I suggest using cookies to save the current value of the click variable, like this (ref):
function createCookie(name, value, days) {
    var expires;
    if (days) {
        var date = new Date();
        date.setTime(date.getTime() + (days * 24 * 60 * 60 * 1000));
        expires = "; expires=" + date.toGMTString();
    }
    else {
        expires = "";
    }
    document.cookie = name + "=" + value + expires + "; path=/";
}
Reading cookies:
function getCookie(c_name) {
    if (document.cookie.length > 0) {
        c_start = document.cookie.indexOf(c_name + "=");
        if (c_start != -1) {
            c_start = c_start + c_name.length + 1;
            c_end = document.cookie.indexOf(";", c_start);
            if (c_end == -1) {
                c_end = document.cookie.length;
            }
            return unescape(document.cookie.substring(c_start, c_end));
        }
    }
    return "";
}
Demo with cookie support: http://jsbin.com/rekafi/4
Note: I suggest you move away from including inline JavaScript in your <button> element (e.g. onClick="incrementClick()"). Try to keep your JS separate from your HTML using jQuery, for example.