Judging by the four Google hits for 'beroepsconclusie', I'd say that it refers to the 'claim on appeal' or, if you like, the 'statement of claim on appeal'. The 'aanvullende beroepsconclusie' would then be a statement outlining an additional claim or claims. I suppose such a statement could even amend the original claims on appeal, which would make it an 'amendment to the statement of claim on appeal' rather than just an addition. Can find no references for this in my dictionaries, however.

-> "Having demanded the reduction of the price, the buyer thereby waived his claim, formulated in his Supplement to the statement of claim, for compensation of expenses in accordance with Article 45, in excess of payment of the amount of reduction of the price."
http://www.cisg.law.pace.edu/cisg/wais/db/cases2/000124r1.ht...

-> "Although Sparks urges another instance of the alleged ineffectiveness oftrial counsel, he did not raise it below. Thus, we will not consider this additional claim on appeal."
http://www2.state.ga.us/Courts/Supreme/pdf/s03a0989.pdf