Challah, Chapter 1, Mishnah 3. This mishnah lists cases in which there is an obligation of challah and yet no obligation of tithes, since the two obligations do not always go hand in hand.
The text of the mishnah:
"Eilu chayavin b'challah u'peturin min hama'asros" - the following are obligated in separating challah, but exempt from separating tithes:
"Haleket v'hashichechah v'hapei'ah v'hahefker" - leket is what falls during the harvesting; shichechah is what the harvester forgot; pei'ah is the corner of the field left standing for the poor and the destitute; and hefker is something that has no owner. All of these are exempt from tithes, and nevertheless they are obligated in challah, since by the time a person makes dough from them he has already acquired them and they have become his, and the obligation of challah takes effect at the time the dough is made.
"U'ma'aser rishon shenitlah terumaso" - first tithe that is given to the Levi, from which terumas ma'aser has already been separated. As first tithe it is exempt from tithes, but the Levi is obligated to separate challah from it.
"U'ma'aser sheni v'hekdesh shenifdu" - second tithe and consecrated produce that have already been redeemed. Even though they were redeemed, they are exempt from tithes, because the redemption took place after the obligation of tithes had already taken effect upon them; the obligation of challah, however, comes later, and therefore they are obligated in it.
"U'mosar ha'omer" - what remains of the omer offering.
"U'tevuah shelo heivi'ah shelish" - grain that has not grown even a third of its full growth.
Explanation of first tithe from which terumah was taken:
The Rishonim explain that this expression, as it is used throughout the Talmud, refers to a case in which terumas ma'aser was separated from it, but terumah gedolah - the original portion given to the kohen - was not separated from it. The case is one of first tithe that was separated early, while the grain was still in its stalks, before the obligation of terumah gedolah took effect, and we learn from the verse that one who separates at this stage is no longer required to separate terumah gedolah from it. Terumas ma'aser, by contrast, does take effect upon it and was separated from it, so it is exempt from tithes but obligated in challah.
Explanation of second tithe and consecrated produce that were redeemed:
According to the Yerushalmi, the case of second tithe as well speaks of one who separated the second tithe early, before the obligation of terumah, and therefore terumah gedolah was never separated from it, and even after the fact he is not obligated to separate it, since it has already been taken as second tithe. And the consecrated produce is exempt for the same reason: it was redeemed after the obligation of tithes, but before the obligation of challah.
Explanation of what remains of the omer:
The omer offering, being new grain, was sifted thoroughly so that it would be fit to be brought in the Beis HaMikdash on the sixteenth of Nissan. As a result, a quantity was left over that was not needed, which still had particles of bran and the like mixed into it, and it left its sanctity and was redeemed. Since at the time the obligation of tithes took effect it was still consecrated, it is exempt from tithes; but at the time the obligation of challah took effect it had already been redeemed, and therefore it is obligated in challah.
Explanation of grain that has not grown a third:
Such grain is exempt from tithes, since regarding tithes the verse says "tevuas zarecha" - the produce of your seed, and from here we learn that it must be "something that is sown and grows" - something that would sprout if it were planted, whereas grain at such an early stage has no capacity to sprout. The criterion for challah, however, is something that can come to leavening, and since this grain can come to leavening even at this early stage, it is obligated in challah.
The view of Rabbi Eliezer:
Rabbi Eliezer disagrees on this last point and holds that "tevuah shelo heivi'ah shelish - peturah min hachallah" - grain that has not grown a third is exempt from challah. He derives this from the verse "ki terumas ha'adamah tihyeh lachem" - for it shall be to you a terumah of the ground, teaching that challah is taken in the same manner in which terumah is taken: just as terumah does not take effect on grain that has not grown a third, so too challah.
In summary: in this mishnah we listed the cases in which there is an obligation of challah without an obligation of tithes: leket, shichechah, pei'ah and hefker; first tithe from which terumah was taken; second tithe and consecrated produce that were redeemed; what remains of the omer; and grain that has not grown a third. The common principle in all of them is the gap between the time the obligation of tithes takes effect and the time the obligation of challah takes effect, and the difference between the criteria of the two obligations, and on the last point Rabbi Eliezer disagreed and derived challah from terumah.