Yaron Yadan raised an interesting point on his Facebook profile. When Samuel searches for the donkeys in Samuel I:9, it is interrupted by the following statement:
ט לְפָנִים בְּיִשְׂרָאֵל, כֹּה-אָמַר הָאִישׁ בְּלֶכְתּוֹ לִדְרוֹשׁ אֱלֹהִים, לְכוּ וְנֵלְכָה, עַד-הָרֹאֶה: כִּי לַנָּבִיא הַיּוֹם, יִקָּרֵא לְפָנִים הָרֹאֶה.
Beforetime in Israel, when a man went to inquire of God, thus he said: 'Come and let us go to the seer'; for he that is now called a prophet (‘navi’ in Hebrew) was beforetime called a seer (‘ro’eh’).
So in Samuel’s time, they used the term ‘navi’. But before that, they called him ‘ro’eh’.
The problem with that is that the word ‘navi’ is mentioned at least twelve times in the Torah (and ro’eh zero times). This appears to indicate that the Torah’s final redaction must have been after or around the time of Samuel. Or, in any case, long after the time of Moses.
A certain Yoav Etzion remarked that there is a verse in this chapter that contradicts a verse in Genesis (Samuel I:10:2):
בְּלֶכְתְּךָ הַיּוֹם, מֵעִמָּדִי, וּמָצָאתָ שְׁנֵי אֲנָשִׁים עִם-קְבֻרַת רָחֵל בִּגְבוּל בִּנְיָמִן, בְּצֶלְצַח; וְאָמְרוּ אֵלֶיךָ, נִמְצְאוּ הָאֲתֹנוֹת אֲשֶׁר הָלַכְתָּ לְבַקֵּשׁ, וְהִנֵּה נָטַשׁ אָבִיךָ אֶת-דִּבְרֵי הָאֲתֹנוֹת, וְדָאַג לָכֶם לֵאמֹר מָה אֶעֱשֶׂה לִבְנִי.
When thou art departed from me to-day, then thou shalt find two men by the tomb of Rachel, in the border of Benjamin at Zelzah; and they will say unto thee: The asses which thou wentest to seek are found; and, lo, thy father hath left off caring for the asses, and is anxious concerning you, saying: What shall I do for my son?
Here, the tomb of Rachel is located within the border of Benjamin. However, Genesis 35:19 mentions that Rachel was buried in Beit Lechem, which is inside of Judah.
In any case, it remains difficult to maintain that the Torah was written during Moses’ or Joshua’s times.