The Cross of Jesus, the very instrument upon which our Salvation was achieved. If there ever was a relic of importance, this would it.
Now there are many places claiming pieces of the True Cross: Notre Dame in Paris, Santa Croce in Rome, Pisa Cathedral, Shaftesbury Abbey in Mediaeval England etc.
This led to much scorn and derision from the Reformers, for instance:
"There is no abbey so poor as not to have a specimen. In some places there are large fragments, as at the Holy Chapel in Paris, at Poitiers, and at Rome, where a good-sized crucifix is said to have been made of it. In brief, if all the pieces that could be found were collected together, they would make a big ship-load. Yet the Gospel testifies that a single man was able to carry it." - Calvin.
So how much is there? It is true it is mostly little slivers, but there are also large pieces. Ethiopia claims to hold a whole wing; Mount Athos and Rome also have large fragments.
Robert de Clari wrote on taking Constantinople that they found pieces of the True Cross "as thick as a man's leg and a fathom accross".
Rohault de Fleury, a 19th century French Architect, tracked down fragments and tried to see how much remained and came to a figure of about 75kg for the cross. He estimated that about a third of a 3-4m cross remained.
This is often claimed to show that the Reformers' claim was wrong.
A few problems:
1. Rohault de Fleury assumed the Cross to be of Pinewood. The four surviving fragments that have been scientifically examined turns out to be Olive. Church tradition however says it was supposed to be made of Cedar, Pine and Cypress based on a passage in Isaiah.
Therefore either many fragments are fake, including most of the best verified ones with Sack of Constantinople pedigrees, or Church tradition is in error.
2. Rohault de Fleury ommited many fragments such as the mentioned Ethiopian one and almost all the fragments dispersed into Northern Europe were destroyed during the Reformation and later bouts of anti-clericalism. He thus severely underestimated the amounts, especcially taking into account destroyed fragments that can be historically verified. We know that Saladin captured a significant piece that has now gone missing, as an example.
3. Rohault de Fleury's book is a 19th century one, so his methods and calculations are somewhat opaque, so it is difficult to verify it as accurate.
4. Few if any have much traceable provenance and often just 'suddenly appear' in an abbey, with no record from where. The most trustworthy ones can be traced to Constantinople, but even here it is very dubious. One could counter that in wars etc. the records were lost, but often why just the records relating to the Cross?
5. The Roman authorities are unlikely to merely discard such a significant piece of timber in such a wood-poor region.
6. There was a thriving trade in relics in the middle ages, often with Charlatans preying on the pious. The Reformers chief claim was the overall 'fakeness' of the relics, which seem somewhat justifiable.
So most fragments are highly dubious and the reformers' criticism carry some weight - especcially if we consider a bit of hyperbole was common.