As said by @HalvarF, there are actually four verbs in the phrase. It's colloquial speech, trying to ask a favour and showing some blush with leaving open a possibility that the request might be denied.
Let's get to the meaning
Kann ich im Laufe der Woche kommen ?
Kann ich mir die ausleihen ?
is concatenated to
Kann ich mir die im Laufe der Woche ausleihen kommen ?
An additional "holen" isn't necessary, leaving it out doesn't change the meaning. But without it the question might be a bit short, depending on the tone could be interpreted in the direction of demanding rather than asking a favour.
A version with more applied grammar (use of Konjunktiv) would be:
Könnte ich im Laufe der Woche kommen um mir die auszuleihen, bitte ?
But don't try this doubling of verbs ("ausleihen holen kommen") with the intention to signal an unsure outcome if you don't know how to use it, it could also be misunderstood as joking.
As to the article, if it is clear from the context which specific things your friend is asking to pick up, then an indefinete article doesn't make sense here. I assume your friend speaks about specific things they want to borrow, and these have a definite article.