Show only the first three tags

From the Codex example at get_the_tags

<?php
$posttags = get_the_tags();
if ($posttags) {
foreach($posttags as $tag) {
echo $tag->name . ' ';
}
}
?>

Or look at get_the_tag_list.

I’m pretty sure that both of those suggestions will display all of the tags.

Sorry missed the goal ??

<?php
$count = 0;
$posttags = get_the_tags();
if ($posttags) {
foreach($posttags as $tag) {
$count++;
if ($count <= 3 ) {
echo $tag->name . ' ';
}
}
}
?>

Is it possible to do this but also make the tags links to the tag archive?

Yes: use get_tag_link().

Where would you use that? Inserting it in place of get_the_tags() breaks the code.

This should do the job:

<?php
$count = 0;
$posttags = get_the_tags();
if ($posttags) {
foreach($posttags as $tag) {
$count++;
if ($count <= 3 ) {
echo '<a href="' . get_tag_link($tag->term_id) . '">' . $tag->name . '</a> ';
}
}
}
?>

Cool. Thank you.